Minimizing atmospheric discharge within a guided surface waveguide probe

ABSTRACT

Disclosed are various embodiments for eliminating or minimizing atmospheric discharge within the internal phasing coil of the guided surface waveguide probe. A guided surface waveguide probe comprises a charge terminal elevated over a lossy conducting medium. The shape of the charge terminal is designed to minimize atmospheric discharge. A top portion of a coil being configured to provide a voltage to the charge terminal with a phase delay that matches a wave tilt angle associated with a complex Brewster angle of incidence associated with the lossy conducting medium is recessed within a hollow region of the charge terminal.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority to and the benefit of co-pending U.S.Provisional application entitled “Minimizing Atmospheric Dischargewithin a Guided Surface Waveguide Probe,” which was filed on Mar. 7,2017 and assigned Application No. 62/467,963, and which is incorporatedby reference in its entirety.

This application is related to co-pending U.S. Non-provisional patentapplication entitled “Excitation and Use of Guided Surface Wave Modes onLossy Media,” which was filed on Mar. 7, 2013 and assigned applicationSer. No. 13/789,538, and was published on Sep. 11, 2014 as PublicationNumber US2014/0252886 A1, and which is incorporated herein by referencein its entirety. This application is also related to co-pending U.S.Non-provisional patent application entitled “Excitation and Use ofGuided Surface Wave Modes on Lossy Media,” which was filed on Mar. 7,2013 and assigned application Ser. No. 13/789,525, and was published onSep. 11, 2014 as Publication Number US2014/0252865 A1, and which isincorporated herein by reference in its entirety. This application isfurther related to co-pending U.S. Non-provisional patent applicationentitled “Excitation and Use of Guided Surface Wave Modes on LossyMedia,” which was filed on Sep. 10, 2014 and assigned application Ser.No. 14/483,089, and which is incorporated herein by reference in itsentirety. This application is further related to co-pending U.S.Non-provisional patent application entitled “Excitation and Use ofGuided Surface Waves,” which was filed on Jun. 2, 2015 and assignedapplication Ser. No. 14/728,492, and which is incorporated herein byreference in its entirety. This application is further related toco-pending U.S. Non-provisional patent application entitled “Excitationand Use of Guided Surface Waves,” which was filed on Jun. 2, 2015 andassigned application Ser. No. 14/728,507, and which is incorporatedherein by reference in its entirety.

BACKGROUND

For over a century, signals transmitted by radio waves involvedradiation fields launched using conventional antenna structures. Incontrast to radio science, electrical power distribution systems in thelast century involved the transmission of energy guided along electricalconductors. This understanding of the distinction between radiofrequency (RF) and power transmission has existed since the early1900's.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the present disclosure can be better understood withreference to the following drawings. The components in the drawings arenot necessarily to scale, emphasis instead being placed upon clearlyillustrating the principles of the disclosure. Moreover, in thedrawings, like reference numerals designate corresponding partsthroughout the several views.

FIG. 1 is a chart that depicts field strength as a function of distancefor a guided electromagnetic field and a radiated electromagnetic field.

FIG. 2 is a drawing that illustrates a propagation interface with tworegions employed for transmission of a guided surface wave according tovarious embodiments of the present disclosure.

FIG. 3 is a drawing that illustrates a guided surface waveguide probedisposed with respect to a propagation interface of FIG. 2 according tovarious embodiments of the present disclosure.

FIG. 4 is a plot of an example of the magnitudes of close-in and far-outasymptotes of first order Hankel functions according to variousembodiments of the present disclosure.

FIGS. 5A and 5B are drawings that illustrate a complex angle ofincidence of an electric field synthesized by a guided surface waveguideprobe according to various embodiments of the present disclosure.

FIG. 6 is a graphical representation illustrating the effect ofelevation of a charge terminal on the location where the electric fieldof FIG. 5A intersects with the lossy conducting medium at a Brewsterangle according to various embodiments of the present disclosure.

FIGS. 7A through 7C are graphical representations of examples of guidedsurface waveguide probes according to various embodiments of the presentdisclosure.

FIGS. 8A through 8C are graphical representations illustrating examplesof equivalent image plane models of the guided surface waveguide probeof FIGS. 3 and 7A-7C according to various embodiments of the presentdisclosure.

FIGS. 9A through 9C are graphical representations illustrating examplesof single-wire transmission line and classic transmission line models ofthe equivalent image plane models of FIGS. 8B and 8C according tovarious embodiments of the present disclosure.

FIG. 9D is a plot illustrating an example of the reactance variation ofa lumped element tank circuit with respect to operating frequencyaccording to various embodiments of the present disclosure.

FIG. 10 is a flow chart illustrating an example of adjusting a guidedsurface waveguide probe of FIGS. 3 and 7A-7C to launch a guided surfacewave along the surface of a lossy conducting medium according to variousembodiments of the present disclosure.

FIG. 11 is a plot illustrating an example of the relationship between awave tilt angle and the phase delay of a guided surface waveguide probeof FIGS. 3 and 7A-7C according to various embodiments of the presentdisclosure.

FIG. 12 is a drawing that illustrates an example of a guided surfacewaveguide probe according to various embodiments of the presentdisclosure.

FIG. 13 is a graphical representation illustrating the incidence of asynthesized electric field at a complex Brewster angle to match theguided surface waveguide mode at the Hankel crossover distance accordingto various embodiments of the present disclosure.

FIG. 14 is a graphical representation of an example of a guided surfacewaveguide probe of FIG. 12 according to various embodiments of thepresent disclosure.

FIG. 15A includes plots of an example of the imaginary and real parts ofa phase delay (Φ_(U)) of a charge terminal T₁ of a guided surfacewaveguide probe according to various embodiments of the presentdisclosure.

FIG. 15B is a schematic diagram of the guided surface waveguide probe ofFIG. 14 according to various embodiments of the present disclosure.

FIG. 16 is a drawing that illustrates an example of a guided surfacewaveguide probe according to various embodiments of the presentdisclosure.

FIG. 17 is a graphical representation of an example of a guided surfacewaveguide probe of FIG. 16 according to various embodiments of thepresent disclosure.

FIGS. 18A through 18C depict examples of receiving structures that canbe employed to receive energy transmitted in the form of a guidedsurface wave launched by a guided surface waveguide probe according tothe various embodiments of the present disclosure.

FIG. 18D is a flow chart illustrating an example of adjusting areceiving structure according to various embodiments of the presentdisclosure.

FIG. 19 depicts an example of an additional receiving structure that canbe employed to receive energy transmitted in the form of a guidedsurface wave launched by a guided surface waveguide probe according tothe various embodiments of the present disclosure.

FIG. 20 illustrates an example guided surface waveguide probe accordingto various embodiments of the present disclosure.

FIG. 21 illustrates the guided surface waveguide probe and substructureof the site shown in FIG. 20 according to various embodiments of thepresent disclosure.

FIG. 22 illustrates the guided surface waveguide probe shown in FIG. 20with an exterior covering according to various embodiments of thepresent disclosure.

FIGS. 23 and 24 illustrate an example of the support structure of theprobe shown in FIG. 20 according to various embodiments of the presentdisclosure.

FIG. 25 is the cross-sectional view A-A designated in FIG. 20 accordingto various embodiments of the present disclosure.

FIG. 26 is the cross-sectional view A-A designated in FIG. 20 andillustrates a number of sections of a coil of the probe according tovarious embodiments of the present disclosure.

FIG. 27 is an enlarged portion of the cross-sectional view A-Adesignated in FIG. 20 according to various embodiments of the presentdisclosure.

FIG. 28 is a cross-sectional view of the charge terminal of the probeshown in FIG. 20 according to various embodiments of the presentdisclosure.

FIGS. 29A and 29B illustrate top and bottom perspective views of a topsupport platform of the probe shown in FIG. 20 according to variousembodiments of the present disclosure.

FIGS. 30 and 31 illustrate various components inside the substructure ofthe probe shown in FIG. 20 according to various embodiments of thepresent disclosure.

FIGS. 32A and 32B illustrate a grounding system of the probe shown inFIG. 20 according to various embodiments of the present disclosure.

FIGS. 33A and 33B illustrate examples of tank circuits of the probeaccording to various embodiments of the present disclosure.

FIG. 34 illustrates an example of a cross-sectional view of the chargeterminal 520 of the guided surface waveguide probe 500 according tovarious embodiments of the present disclosure.

FIG. 35 illustrates an example of a portion of another embodiment of aguided surface waveguide probe according to various embodiments of thepresent disclosure.

FIG. 36 illustrates an example of a portion of another embodiment of theguided surface waveguide probe of FIG. 35 according to variousembodiments of the present disclosure.

DETAILED DESCRIPTION

To begin, some terminology shall be established to provide clarity inthe discussion of concepts to follow. First, as contemplated herein, aformal distinction is drawn between radiated electromagnetic fields andguided electromagnetic fields.

As contemplated herein, a radiated electromagnetic field compriseselectromagnetic energy that is emitted from a source structure in theform of waves that are not bound to a waveguide. For example, a radiatedelectromagnetic field is generally a field that leaves an electricstructure such as an antenna and propagates through the atmosphere orother medium and is not bound to any waveguide structure. Once radiatedelectromagnetic waves leave an electric structure such as an antenna,they continue to propagate in the medium of propagation (such as air)independent of their source until they dissipate regardless of whetherthe source continues to operate. Once electromagnetic waves areradiated, they are not recoverable unless intercepted, and, if notintercepted, the energy inherent in the radiated electromagnetic wavesis lost forever. Electrical structures such as antennas are designed toradiate electromagnetic fields by maximizing the ratio of the radiationresistance to the structure loss resistance. Radiated energy spreads outin space and is lost regardless of whether a receiver is present. Theenergy density of the radiated fields is a function of distance due togeometric spreading. Accordingly, the term “radiate” in all its forms asused herein refers to this form of electromagnetic propagation.

A guided electromagnetic field is a propagating electromagnetic wavewhose energy is concentrated within or near boundaries between mediahaving different electromagnetic properties. In this sense, a guidedelectromagnetic field is one that is bound to a waveguide and may becharacterized as being conveyed by the current flowing in the waveguide.If there is no load to receive and/or dissipate the energy conveyed in aguided electromagnetic wave, then no energy is lost except for thatwhich is dissipated in the conductivity of the guiding medium. Statedanother way, if there is no load for a guided electromagnetic wave, thenno energy is consumed. Thus, a generator or other source generating aguided electromagnetic field does not deliver real power unless aresistive load is present. To this end, such a generator or other sourceessentially runs idle until a load is presented. This is akin to runninga generator to generate a 60 Hertz electromagnetic wave that istransmitted over power lines where there is no electrical load. Itshould be noted that a guided electromagnetic field or wave is theequivalent to what is termed a “transmission line mode.” This contrastswith radiated electromagnetic waves in which real power is supplied atall times in order to generate radiated waves. Unlike radiatedelectromagnetic waves, guided electromagnetic energy does not continueto propagate along a finite length waveguide after the energy source isturned off. Accordingly, the term “guide” in all its forms as usedherein refers to this transmission mode of electromagnetic propagation.

Referring now to FIG. 1, shown is a graph 100 of field strength indecibels (dB) above an arbitrary reference in volts per meter as afunction of distance in kilometers on a log-dB plot to furtherillustrate the distinction between radiated and guided electromagneticfields. The graph 100 of FIG. 1 depicts a guided field strength curve103 that shows the field strength of a guided electromagnetic field as afunction of distance. This guided field strength curve 103 isessentially the same as a transmission line mode. Also, the graph 100 ofFIG. 1 depicts a radiated field strength curve 106 that shows the fieldstrength of a radiated electromagnetic field as a function of distance.

Of interest are the shapes of the curves 103 and 106 for guided wave andfor radiation propagation, respectively. The radiated field strengthcurve 106 falls off geometrically (1/d, where d is distance), which isdepicted as a straight line on the log-log scale. The guided fieldstrength curve 103, on the other hand, has a characteristic exponentialdecay of e^(−αd)/√{square root over (d)} and exhibits a distinctive knee109 on the log-log scale. The guided field strength curve 103 and theradiated field strength curve 106 intersect at point 112, which occursat a crossing distance. At distances less than the crossing distance atintersection point 112, the field strength of a guided electromagneticfield is significantly greater at most locations than the field strengthof a radiated electromagnetic field. At distances greater than thecrossing distance, the opposite is true. Thus, the guided and radiatedfield strength curves 103 and 106 further illustrate the fundamentalpropagation difference between guided and radiated electromagneticfields. For an informal discussion of the difference between guided andradiated electromagnetic fields, reference is made to Milligan, T.,Modern Antenna Design, McGraw-Hill, 1^(st) Edition, 1985, pp. 8-9, whichis incorporated herein by reference in its entirety.

The distinction between radiated and guided electromagnetic waves, madeabove, is readily expressed formally and placed on a rigorous basis.That two such diverse solutions could emerge from one and the samelinear partial differential equation, the wave equation, analyticallyfollows from the boundary conditions imposed on the problem. The Greenfunction for the wave equation, itself, contains the distinction betweenthe nature of radiation and guided waves.

In empty space, the wave equation is a differential operator whoseeigenfunctions possess a continuous spectrum of eigenvalues on thecomplex wave-number plane. This transverse electro-magnetic (TEM) fieldis called the radiation field, and those propagating fields are called“Hertzian waves.” However, in the presence of a conducting boundary, thewave equation plus boundary conditions mathematically lead to a spectralrepresentation of wave-numbers composed of a continuous spectrum plus asum of discrete spectra. To this end, reference is made to Sommerfeld,A., “Uber die Ausbreitung der Wellen in der Drahtlosen Telegraphie,”Annalen der Physik, Vol. 28, 1909, pp. 665-736. Also see Sommerfeld, A.,“Problems of Radio,” published as Chapter 6 in Partial DifferentialEquations in Physics—Lectures on Theoretical Physics: Volume VI,Academic Press, 1949, pp. 236-289, 295-296; Collin, R. E., “HertzianDipole Radiating Over a Lossy Earth or Sea: Some Early and Late 20^(th)Century Controversies,” IEEE Antennas and Propagation Magazine, Vol. 46,No. 2, April 2004, pp. 64-79; and Reich, H. J., Ordnung, P. F, Krauss,H. L., and Skalnik, J. G., Microwave Theory and Techniques, VanNostrand, 1953, pp. 291-293, each of these references being incorporatedherein by reference in its entirety.

The terms “ground wave” and “surface wave” identify two distinctlydifferent physical propagation phenomena. A surface wave arisesanalytically from a distinct pole yielding a discrete component in theplane wave spectrum. See, e.g., “The Excitation of Plane Surface Waves”by Cullen, A. L., (Proceedings of the IEE (British), Vol. 101, Part IV,August 1954, pp. 225-235). In this context, a surface wave is consideredto be a guided surface wave. The surface wave (in the Zenneck-Sommerfeldguided wave sense) is, physically and mathematically, not the same asthe ground wave (in the Weyl-Norton-FCC sense) that is now so familiarfrom radio broadcasting. These two propagation mechanisms arise from theexcitation of different types of eigenvalue spectra (continuum ordiscrete) on the complex plane. The field strength of the guided surfacewave decays exponentially with distance as illustrated by guided fieldstrength curve 103 of FIG. 1 (much like propagation in a lossywaveguide) and resembles propagation in a radial transmission line, asopposed to the classical Hertzian radiation of the ground wave, whichpropagates spherically, possesses a continuum of eigenvalues, falls offgeometrically as illustrated by radiated field strength curve 106 ofFIG. 1, and results from branch-cut integrals. As experimentallydemonstrated by C. R. Burrows in “The Surface Wave in Radio Propagationover Plane Earth” (Proceedings of the IRE, Vol. 25, No. 2, February,1937, pp. 219-229) and “The Surface Wave in Radio Transmission” (BellLaboratories Record, Vol. 15, June 1937, pp. 321-324), vertical antennasradiate ground waves but do not launch guided surface waves.

To summarize the above, first, the continuous part of the wave-numbereigenvalue spectrum, corresponding to branch-cut integrals, produces theradiation field, and second, the discrete spectra, and correspondingresidue sum arising from the poles enclosed by the contour ofintegration, result in non-TEM traveling surface waves that areexponentially damped in the direction transverse to the propagation.Such surface waves are guided transmission line modes. For furtherexplanation, reference is made to Friedman, B., Principles andTechniques of Applied Mathematics, Wiley, 1956, pp. pp. 214, 283-286,290, 298-300.

In free space, antennas excite the continuum eigenvalues of the waveequation, which is a radiation field, where the outwardly propagating RFenergy with E_(z) and H_(ϕ) in-phase is lost forever. On the other hand,waveguide probes excite discrete eigenvalues, which results intransmission line propagation. See Collin, R. E., Field Theory of GuidedWaves, McGraw-Hill, 1960, pp. 453, 474-477. While such theoreticalanalyses have held out the hypothetical possibility of launching opensurface guided waves over planar or spherical surfaces of lossy,homogeneous media, for more than a century no known structures in theengineering arts have existed for accomplishing this with any practicalefficiency. Unfortunately, since it emerged in the early 1900's, thetheoretical analysis set forth above has essentially remained a theoryand there have been no known structures for practically accomplishingthe launching of open surface guided waves over planar or sphericalsurfaces of lossy, homogeneous media.

According to the various embodiments of the present disclosure, variousguided surface waveguide probes are described that are configured toexcite electric fields that couple into a guided surface waveguide modealong the surface of a lossy conducting medium. Such guidedelectromagnetic fields are substantially mode-matched in magnitude andphase to a guided surface wave mode on the surface of the lossyconducting medium. Such a guided surface wave mode can also be termed aZenneck waveguide mode. By virtue of the fact that the resultant fieldsexcited by the guided surface waveguide probes described herein aresubstantially mode-matched to a guided surface waveguide mode on thesurface of the lossy conducting medium, a guided electromagnetic fieldin the form of a guided surface wave is launched along the surface ofthe lossy conducting medium. According to one embodiment, the lossyconducting medium comprises a terrestrial medium such as the Earth.

Referring to FIG. 2, shown is a propagation interface that provides foran examination of the boundary value solutions to Maxwell's equationsderived in 1907 by Jonathan Zenneck as set forth in his paper Zenneck,J., “On the Propagation of Plane Electromagnetic Waves Along a FlatConducting Surface and their Relation to Wireless Telegraphy,” Annalender Physik, Serial 4, Vol. 23, Sep. 20, 1907, pp. 846-866. FIG. 2depicts cylindrical coordinates for radially propagating waves along theinterface between a lossy conducting medium specified as Region 1 and aninsulator specified as Region 2. Region 1 can comprise, for example, anylossy conducting medium. In one example, such a lossy conducting mediumcan comprise a terrestrial medium such as the Earth or other medium.Region 2 is a second medium that shares a boundary interface with Region1 and has different constitutive parameters relative to Region 1. Region2 can comprise, for example, any insulator such as the atmosphere orother medium. The reflection coefficient for such a boundary interfacegoes to zero only for incidence at a complex Brewster angle. SeeStratton, J. A., Electromagnetic Theory, McGraw-Hill, 1941, p. 516.

According to various embodiments, the present disclosure sets forthvarious guided surface waveguide probes that generate electromagneticfields that are substantially mode-matched to a guided surface waveguidemode on the surface of the lossy conducting medium comprising Region 1.According to various embodiments, such electromagnetic fieldssubstantially synthesize a wave front incident at a complex Brewsterangle of the lossy conducting medium that can result in zero reflection.

To explain further, in Region 2, where an e^(jωt) field variation isassumed and where ρ≠0 and z≥0 (with z being the vertical coordinatenormal to the surface of Region 1, and ρ being the radial dimension incylindrical coordinates), Zenneck's closed-form exact solution ofMaxwell's equations satisfying the boundary conditions along theinterface are expressed by the following electric field and magneticfield components:

$\begin{matrix}{{H_{2\; \varphi} = {{Ae}^{{- u_{2}}z}\mspace{14mu} {H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}},} & (1) \\{{E_{2\; \rho} = {{A\left( \frac{u_{2}}{j\; \omega \; ɛ_{o}} \right)}e^{{- u_{2}}z}\mspace{14mu} {H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}},{and}} & (2) \\{E_{2z} = {{A\left( \frac{- \gamma}{\omega \; ɛ_{o}} \right)}e^{{- u_{2}}z}\mspace{14mu} {{H_{0}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}.}}} & (3)\end{matrix}$

In Region 1, where the e^(jωt) field variation is assumed and where ρ ≠0and z≤0, Zenneck's closed-form exact solution of Maxwell's equationssatisfying the boundary conditions along the interface is expressed bythe following electric field and magnetic field components:

$\begin{matrix}{{H_{1\; \varphi} = {{Ae}^{u_{1}z}\mspace{14mu} {H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}},} & (4) \\{{{E_{1\; \rho} = {{A\left( \frac{- u_{1}}{\sigma_{1} + {j\; \omega \; ɛ_{1}}} \right)}e^{u_{1}z}\mspace{14mu} {H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}},{and}}\mspace{191mu}} & {\; (5)} \\{E_{1z} = {{A\left( \frac{{- j}\; \gamma}{\sigma_{1} + {j\; \omega \; ɛ_{1}}} \right)}e^{u_{1}z}\mspace{14mu} {{H_{0}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}.}}} & (6)\end{matrix}$

In these expressions, z is the vertical coordinate normal to the surfaceof Region 1 and p is the radial coordinate, H_(n) ⁽²⁾(−jγρ) is a complexargument Hankel function of the second kind and order n, u₁ is thepropagation constant in the positive vertical (z) direction in Region 1,u₂ is the propagation constant in the vertical (z) direction in Region2, σ₁ is the conductivity of Region 1, ω is equal to 2πf, where f is afrequency of excitation, ε_(o) is the permittivity of free space, ε₁ isthe permittivity of Region 1, A is a source constant imposed by thesource, and γ is a surface wave radial propagation constant.

The propagation constants in the ±z directions are determined byseparating the wave equation above and below the interface betweenRegions 1 and 2, and imposing the boundary conditions. This exercisegives, in Region 2,

$\begin{matrix}{u_{2} = \frac{- {jk}_{o}}{\sqrt{1 + \left( {ɛ_{r} - {jx}} \right)}}} & (7)\end{matrix}$

and gives, in Region 1,

u ₁ =−u ₂(ε_(r) −jx).  (8)

The radial propagation constant γ is given by

$\begin{matrix}{{\gamma = {{j\sqrt{k_{o}^{2} + u_{2}^{2}}} = {j\frac{k_{o}n}{\sqrt{1 + n^{2}}}}}},} & (9)\end{matrix}$

which is a complex expression where n is the complex index of refractiongiven by

n=√{square root over (ε_(r) −jx)}.  (10)

In all of the above Equations,

$\begin{matrix}{{x = \frac{\sigma_{1}}{\omega \; ɛ_{o}}},{and}} & (11) \\{{k_{o} = {{\omega \sqrt{\mu_{o}ɛ_{o}}} = \frac{\lambda_{o}}{2\; \pi}}},} & (12)\end{matrix}$

where ε_(r) comprises the relative permittivity of Region 1, σ₁ is theconductivity of Region 1, ε_(o) is the permittivity of free space, andμ_(o), comprises the permeability of free space. Thus, the generatedsurface wave propagates parallel to the interface and exponentiallydecays vertical to it. This is known as evanescence.

Thus, Equations (1)-(3) can be considered to be acylindrically-symmetric, radially-propagating waveguide mode. SeeBarlow, H. M., and Brown, J., Radio Surface Waves, Oxford UniversityPress, 1962, pp. 10-12, 29-33. The present disclosure details structuresthat excite this “open boundary” waveguide mode. Specifically, accordingto various embodiments, a guided surface waveguide probe is providedwith a charge terminal of appropriate size that is fed with voltageand/or current and is positioned relative to the boundary interfacebetween Region 2 and Region 1. This may be better understood withreference to FIG. 3, which shows an example of a guided surfacewaveguide probe 200 a that includes a charge terminal T₁ elevated abovea lossy conducting medium 203 (e.g., the Earth) along a vertical axis zthat is normal to a plane presented by the lossy conducting medium 203.The lossy conducting medium 203 makes up Region 1, and a second medium206 makes up Region 2 and shares a boundary interface with the lossyconducting medium 203.

According to one embodiment, the lossy conducting medium 203 cancomprise a terrestrial medium such as the planet Earth. To this end,such a terrestrial medium comprises all structures or formationsincluded thereon whether natural or man-made. For example, such aterrestrial medium can comprise natural elements such as rock, soil,sand, fresh water, sea water, trees, vegetation, and all other naturalelements that make up our planet. In addition, such a terrestrial mediumcan comprise man-made elements such as concrete, asphalt, buildingmaterials, and other man-made materials. In other embodiments, the lossyconducting medium 203 can comprise some medium other than the Earth,whether naturally occurring or man-made. In other embodiments, the lossyconducting medium 203 can comprise other media such as man-made surfacesand structures such as automobiles, aircraft, man-made materials (suchas plywood, plastic sheeting, or other materials) or other media.

In the case where the lossy conducting medium 203 comprises aterrestrial medium or Earth, the second medium 206 can comprise theatmosphere above the ground. As such, the atmosphere can be termed an“atmospheric medium” that comprises air and other elements that make upthe atmosphere of the Earth. In addition, it is possible that the secondmedium 206 can comprise other media relative to the lossy conductingmedium 203.

The guided surface waveguide probe 200 a includes a feed network 209that couples an excitation source 212 to the charge terminal T₁ via,e.g., a vertical feed line conductor. The excitation source 212 maycomprise, for example, an Alternating Current (AC) source or some othersource. As contemplated herein, an excitation source can comprise an ACsource or other type of source. According to various embodiments, acharge Q₁ is imposed on the charge terminal T₁ to synthesize an electricfield based upon the voltage applied to terminal T₁ at any giveninstant. Depending on the angle of incidence (θ_(i)) of the electricfield (E), it is possible to substantially mode-match the electric fieldto a guided surface waveguide mode on the surface of the lossyconducting medium 203 comprising Region 1.

By considering the Zenneck closed-form solutions of Equations (1)-(6),the Leontovich impedance boundary condition between Region 1 and Region2 can be stated as

{circumflex over (z)}×

₂(ρ,φ,0)=

_(S),  (13)

where {circumflex over (z)} is a unit normal in the positive vertical(+z) direction and

₂ is the magnetic field strength in Region 2 expressed by Equation (1)above. Equation (13) implies that the electric and magnetic fieldsspecified in Equations (1)-(3) may result in a radial surface currentdensity along the boundary interface, where the radial surface currentdensity can be specified by

J _(ρ)(ρ′)=−A H ₁ ⁽²⁾(−jγρ′)  (14)

where A is a constant. Further, it should be noted that close-in to theguided surface waveguide probe 200 (for ρ<<λ), Equation (14) above hasthe behavior

$\begin{matrix}{{J_{close}\left( \rho^{\prime} \right)} = {\frac{- {A\left( {j\; 2} \right)}}{\pi \left( {{- j}\; \gamma \; \rho^{\prime}} \right)} = {{- H_{\varphi}} = {- {\frac{I_{o}}{2\; \pi \; \rho^{\prime}}.}}}}} & (15)\end{matrix}$

The negative sign means that when source current (I_(o)) flowsvertically upward as illustrated in FIG. 3, the “close-in” groundcurrent flows radially inward. By field matching on H_(ϕ) “close-in,” itcan be determined that

$\begin{matrix}{A = {{- \frac{I_{o}\gamma}{4}} = {- \frac{\omega \; q_{1}\gamma}{4}}}} & (16)\end{matrix}$

where q₁=C₁V₁, in Equations (1)-(6) and (14). Therefore, the radialsurface current density of Equation (14) can be restated as

$\begin{matrix}{{J_{\rho}\left( \rho^{\prime} \right)} = {\frac{I_{o}\gamma}{4}\; {{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho^{\prime}} \right)}.}}} & (17)\end{matrix}$

The fields expressed by Equations (1)-(6) and (17) have the nature of atransmission line mode bound to a lossy interface, not radiation fieldsthat are associated with groundwave propagation. See Barlow, H. M. andBrown, J., Radio Surface Waves, Oxford University Press, 1962, pp. 1-5.

At this point, a review of the nature of the Hankel functions used inEquations (1)-(6) and (17) is provided for these solutions of the waveequation. One might observe that the Hankel functions of the first andsecond kind and order n are defined as complex combinations of thestandard Bessel functions of the first and second kinds

H _(n) ⁽¹⁾(x)=J _(n)(x)+jN _(n)(x), and  (18)

H _(n) ⁽²⁾(x)=J _(n)(x)−jN _(n)(x).  (19)

These functions represent cylindrical waves propagating radially inward(H_(n) ⁽¹⁾) and outward (H_(n) ⁽²⁾), respectively. The definition isanalogous to the relationship e^(±jx)=cos x±j sin x. See, for example,Harrington, R. F., Time-Harmonic Fields, McGraw-Hill, 1961, pp. 460-463.

That H_(n) ⁽²⁾(k_(ρ)ρ) is an outgoing wave can be recognized from itslarge argument asymptotic behavior that is obtained directly from theseries definitions of J_(n)(x) and N_(n)(x). Far-out from the guidedsurface waveguide probe:

$\begin{matrix}{{{{H_{n}^{(2)}(x)}\underset{x\rightarrow\infty}{\rightarrow}{\sqrt{\frac{2j}{\pi \; x}}j^{n}e^{- {jx}}}} = {\sqrt{\frac{2}{\pi \; x}}j^{n}e^{- {j{({x - \frac{\pi}{4}})}}}}},} & \left( {20a} \right)\end{matrix}$

which, when multiplied by e^(jωt), is an outward propagating cylindricalwave of the form e^(j(ωt−kρ)) with a 1/√{square root over (ρ)} spatialvariation. The first order (n=1) solution can be determined fromEquation (20a) to be

$\begin{matrix}{{{H_{1}^{(2)}(x)}\underset{x\rightarrow\infty}{\rightarrow}{j\sqrt{\frac{2j}{\pi \; x}}e^{- {jx}}}} = {\sqrt{\frac{2}{\pi \; x}}\; {e^{- {j{({x - \frac{\pi}{2} - \frac{\pi}{4}})}}}.}}} & \left( {20b} \right)\end{matrix}$

Close-in to the guided surface waveguide probe (for ρ<<λ), the Hankelfunction of first order and the second kind behaves as

$\begin{matrix}{{H_{1}^{(2)}(x)}\underset{x\rightarrow 0}{\rightarrow}\; {\frac{2j}{\pi \; x}.}} & (21)\end{matrix}$

Note that these asymptotic expressions are complex quantities. When x isa real quantity, Equations (20b) and (21) differ in phase by √{squareroot over (j)}, which corresponds to an extra phase advance or “phaseboost” of 45° or, equivalently, λ/8. The close-in and far-out asymptotesof the first order Hankel function of the second kind have a Hankel“crossover” or transition point where they are of equal magnitude at adistance of ρ=R_(x).

Thus, beyond the Hankel crossover point the “far out” representationpredominates over the “close-in” representation of the Hankel function.The distance to the Hankel crossover point (or Hankel crossoverdistance) can be found by equating Equations (20b) and (21) for −jγρ,and solving for R_(x). With x=σ/ωε_(o), it can be seen that the far-outand close-in Hankel function asymptotes are frequency dependent, withthe Hankel crossover point moving out as the frequency is lowered. Itshould also be noted that the Hankel function asymptotes may also varyas the conductivity (σ) of the lossy conducting medium changes. Forexample, the conductivity of the soil can vary with changes in weatherconditions.

Referring to FIG. 4, shown is an example of a plot of the magnitudes ofthe first order Hankel functions of Equations (20b) and (21) for aRegion 1 conductivity of σ=0.010 mhos/m and relative permittivityε_(r)=15, at an operating frequency of 1850 kHz. Curve 115 is themagnitude of the far-out asymptote of Equation (20b) and curve 118 isthe magnitude of the close-in asymptote of Equation (21), with theHankel crossover point 121 occurring at a distance of R_(x)=54 feet.While the magnitudes are equal, a phase offset exists between the twoasymptotes at the Hankel crossover point 121. It can also be seen thatthe Hankel crossover distance is much less than a wavelength of theoperation frequency.

Considering the electric field components given by Equations (2) and (3)of the Zenneck closed-form solution in Region 2, it can be seen that theratio of E_(z) and E_(ρ) asymptotically passes to

$\begin{matrix}{{\frac{E_{z}}{E_{\rho}} = {{{\left( \frac{{- j}\; \gamma}{u_{2}} \right)\frac{H_{0}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}\underset{\rho\rightarrow\infty}{\rightarrow}\; \sqrt{ɛ_{r} - {j\frac{\sigma}{\omega \; ɛ_{o}}}}} = {n = {\tan \; \theta_{i}}}}},} & (22)\end{matrix}$

where n is the complex index of refraction of Equation (10) and θ_(i) isthe angle of incidence of the electric field. In addition, the verticalcomponent of the mode-matched electric field of Equation (3)asymptotically passes to

$\begin{matrix}{{E_{2\; z}\underset{\rho\rightarrow\infty}{\rightarrow}{\left( \frac{q_{free}}{ɛ_{o}} \right)\sqrt{\frac{\gamma^{3}}{8\pi}}e^{{- u_{2}}z}\frac{e^{- {j{({{\gamma\rho} - {\pi/4}})}}}}{\sqrt{\rho}}}},} & (23)\end{matrix}$

which is linearly proportional to free charge on the isolated componentof the elevated charge terminal's capacitance at the terminal voltage,q_(free)=C_(free)×V_(T).

For example, the height H₁ of the elevated charge terminal T₁ in FIG. 3affects the amount of free charge on the charge terminal T₁. When thecharge terminal T₁ is near the ground plane of Region 1, most of thecharge Q₁ on the terminal is “bound.” As the charge terminal T₁ iselevated, the bound charge is lessened until the charge terminal T₁reaches a height at which substantially all of the isolated charge isfree.

The advantage of an increased capacitive elevation for the chargeterminal T₁ is that the charge on the elevated charge terminal T₁ isfurther removed from the ground plane, resulting in an increased amountof free charge q_(free) to couple energy into the guided surfacewaveguide mode. As the charge terminal T₁ is moved away from the groundplane, the charge distribution becomes more uniformly distributed aboutthe surface of the terminal. The amount of free charge is related to theself-capacitance of the charge terminal T₁.

For example, the capacitance of a spherical terminal can be expressed asa function of physical height above the ground plane. The capacitance ofa sphere at a physical height of h above a perfect ground is given by

C _(elevated sphere)=4πε_(o) a(1+M+M ² +M ³+2M ⁴+3M ⁵+ . . . ),  (24)

where the diameter of the sphere is 2a, and where M=a/2h with h beingthe height of the spherical terminal. As can be seen, an increase in theterminal height h reduces the capacitance C of the charge terminal. Itcan be shown that for elevations of the charge terminal T₁ that are at aheight of about four times the diameter (4D=8a) or greater, the chargedistribution is approximately uniform about the spherical terminal,which can improve the coupling into the guided surface waveguide mode.

In the case of a sufficiently isolated terminal, the self-capacitance ofa conductive sphere can be approximated by C=47πε_(o)a, where a is theradius of the sphere in meters, and the self-capacitance of a disk canbe approximated by C=8ε_(o)a, where a is the radius of the disk inmeters. The charge terminal T₁ can include any shape such as a sphere, adisk, a cylinder, a cone, a torus, a hood, one or more rings, or anyother randomized shape or combination of shapes. An equivalent sphericaldiameter can be determined and used for positioning of the chargeterminal T₁.

This may be further understood with reference to the example of FIG. 3,where the charge terminal T₁ is elevated at a physical height ofh_(p)=H₁ above the lossy conducting medium 203. To reduce the effects ofthe “bound” charge, the charge terminal T₁ can be positioned at aphysical height that is at least four times the spherical diameter (orequivalent spherical diameter) of the charge terminal T₁ to reduce thebounded charge effects.

Referring next to FIG. 5A, shown is a ray optics interpretation of theelectric field produced by the elevated charge Q₁ on charge terminal T₁of FIG. 3. As in optics, minimizing the reflection of the incidentelectric field can improve and/or maximize the energy coupled into theguided surface waveguide mode of the lossy conducting medium 203. For anelectric field (E_(∥)) that is polarized parallel to the plane ofincidence (not the boundary interface), the amount of reflection of theincident electric field may be determined using the Fresnel reflectioncoefficient, which can be expressed as

$\begin{matrix}{{{\Gamma_{}\left( \theta_{i} \right)} = {\frac{E_{{},R}}{E_{{},i}} = \frac{\sqrt{\left( {ɛ_{r} - {jx}} \right) - {\sin^{2}\theta_{i}}} - {\left( {ɛ_{r} - {jx}} \right)\cos \; \theta_{i}}}{\sqrt{\left( {ɛ_{r} - {jx}} \right) - {\sin^{2}\theta_{i}}} + {\left( {ɛ_{r} - {jx}} \right)\cos \; \theta_{i}}}}},} & (25)\end{matrix}$

where θ_(i) is the conventional angle of incidence measured with respectto the surface normal.

In the example of FIG. 5A, the ray optic interpretation shows theincident field polarized parallel to the plane of incidence having anangle of incidence of θ_(i), which is measured with respect to thesurface normal (2). There will be no reflection of the incident electricfield when Γ_(∥)(θ_(i))=0 and thus the incident electric field will becompletely coupled into a guided surface waveguide mode along thesurface of the lossy conducting medium 203. It can be seen that thenumerator of Equation (25) goes to zero when the angle of incidence is

θ_(i)=arctan(√{square root over (ε_(r) −jx)})=θ_(i,B),  (26)

where x=σ/ωε_(o). This complex angle of incidence (θ_(i,B)) is referredto as the Brewster angle. Referring back to Equation (22), it can beseen that the same complex Brewster angle (θ_(i,B)) relationship ispresent in both Equations (22) and (26).

As illustrated in FIG. 5A, the electric field vector E can be depictedas an incoming non-uniform plane wave, polarized parallel to the planeof incidence. The electric field vector E can be created fromindependent horizontal and vertical components as

(θ_(i))=E _(ρ) {circumflex over (ρ)}+E _(z) {circumflex over (z)}.  (27)

Geometrically, the illustration in FIG. 5A suggests that the electricfield vector E can be given by

$\begin{matrix}{{{E_{\rho}\left( {\rho,z} \right)} = {{E\left( {\rho,z} \right)}\cos \; \theta_{i}}},{and}} & \left( {28a} \right) \\{{{E_{z}\left( {\rho,z} \right)} = {{{E\left( {\rho,z} \right)}{\cos \left( {\frac{\pi}{2} - \theta_{i}} \right)}} = {{E\left( {\rho,z} \right)}\sin \; \theta_{i}}}},} & \left( {28b} \right)\end{matrix}$

which means that the field ratio is

$\begin{matrix}{\frac{E_{\rho}}{E_{z}} = {\frac{1}{\tan \; \theta_{i}} = {\tan \; {\psi_{i}.}}}} & (29)\end{matrix}$

A generalized parameter W, called “wave tilt,” is noted herein as theratio of the horizontal electric field component to the verticalelectric field component given by

$\begin{matrix}{{W = {\frac{E_{\rho}}{E_{z}} = {{W}e^{j\; \Psi}}}},{or}} & \left( {30a} \right) \\{{\frac{1}{W} = {\frac{E_{z}}{E_{\rho}} = {{\tan \; \theta_{i}} = {\frac{1}{W}e^{{- j}\; \Psi}}}}},} & \left( {30b} \right)\end{matrix}$

which is complex and has both magnitude and phase. For anelectromagnetic wave in Region 2 (FIG. 2), the wave tilt angle (Ψ) isequal to the angle between the normal of the wave-front at the boundaryinterface with Region 1 (FIG. 2) and the tangent to the boundaryinterface. This may be easier to see in FIG. 5B, which illustratesequi-phase surfaces of an electromagnetic wave and their normals for aradial cylindrical guided surface wave. At the boundary interface (z=0)with a perfect conductor, the wave-front normal is parallel to thetangent of the boundary interface, resulting in W=0. However, in thecase of a lossy dielectric, a wave tilt W exists because the wave-frontnormal is not parallel with the tangent of the boundary interface atz=0.

Applying Equation (30b) to a guided surface wave gives

$\begin{matrix}{{\tan \; \theta_{i,B}} = {\frac{E_{z}}{E_{\rho}} = {\frac{u_{2}}{\gamma} = {\sqrt{ɛ_{r} - {jx}} = {n = {\frac{1}{W} = {\frac{1}{W}{e^{{- j}\; \Psi}.}}}}}}}} & (31)\end{matrix}$

With the angle of incidence equal to the complex Brewster angle(θ_(i,B)), the Fresnel reflection coefficient of Equation (25) vanishes,as shown by

$\begin{matrix}{{\Gamma_{}\left( \theta_{i,B} \right)} = {\left. \frac{\sqrt{\left( {ɛ_{r} - {jx}} \right) - {\sin^{2}\theta_{i}}} - {\left( {ɛ_{r} - {jx}} \right)\cos \; \theta_{i}}}{\sqrt{\left( {ɛ_{r} - {jx}} \right) - {\sin^{2}\theta_{i}}} + {\left( {ɛ_{r} - {jx}} \right)\cos \; \theta_{i}}} \right|_{\theta_{i} = \theta_{i,B}} = 0.}} & (32)\end{matrix}$

By adjusting the complex field ratio of Equation (22), an incident fieldcan be synthesized to be incident at a complex angle at which thereflection is reduced or eliminated. Establishing this ratio asn=√{square root over (ε_(r)−jx)} results in the synthesized electricfield being incident at the complex Brewster angle, making thereflections vanish.

The concept of an electrical effective height can provide furtherinsight into synthesizing an electric field with a complex angle ofincidence with a guided surface waveguide probe 200. The electricaleffective height (h_(eff)) has been defined as

$\begin{matrix}{H_{eff} = {\frac{1}{I_{0}}{\int_{0}^{h_{p}}{{I(z)}{dz}}}}} & (33)\end{matrix}$

for a monopole with a physical height (or length) of h_(p). Since theexpression depends upon the magnitude and phase of the sourcedistribution along the structure, the effective height (or length) iscomplex in general. The integration of the distributed current I(z) ofthe structure is performed over the physical height of the structure(h_(p)), and normalized to the ground current (I₀) flowing upwardthrough the base (or input) of the structure. The distributed currentalong the structure can be expressed by

I(z)=I _(C) cos(β₀ z),  (34)

where β₀ is the propagation factor for current propagating on thestructure. In the example of FIG. 3, I_(C) is the current that isdistributed along the vertical structure of the guided surface waveguideprobe 200 a.

For example, consider a feed network 209 that includes a low loss coil(e.g., a helical coil) at the bottom of the structure and a verticalfeed line conductor connected between the coil and the charge terminalT₁. The phase delay due to the coil (or helical delay line) isθ_(c)=β_(p)l_(C), with a physical length of l_(C) and a propagationfactor of

$\begin{matrix}{{\beta_{p} = {\frac{2\pi}{\lambda_{p}} = \frac{2\pi}{V_{f}\lambda_{0}}}},} & (35)\end{matrix}$

where V_(f) is the velocity factor on the structure, λ₀ is thewavelength at the supplied frequency, and λ_(p) is the propagationwavelength resulting from the velocity factor V_(f). The phase delay ismeasured relative to the ground (stake or system) current I₀.

In addition, the spatial phase delay along the length l_(w) of thevertical feed line conductor can be given by θ_(y)=β_(w)l_(w) whereβ_(w) is the propagation phase constant for the vertical feed lineconductor. In some implementations, the spatial phase delay may beapproximated by θ_(y)=β_(w)h_(p), since the difference between thephysical height h_(p) of the guided surface waveguide probe 200 a andthe vertical feed line conductor length l_(W) is much less than awavelength at the supplied frequency (λ₀). As a result, the total phasedelay through the coil and vertical feed line conductor isω=θ_(c)+θ_(y), and the current fed to the top of the coil from thebottom of the physical structure is

I _(C)(θ_(c)+θ_(y))=I ₀ e ^(jΦ),  (36)

with the total phase delay Φ measured relative to the ground (stake orsystem) current I₀. Consequently, the electrical effective height of aguided surface waveguide probe 200 can be approximated by

$\begin{matrix}{{h_{eff} = {{\frac{1}{I_{0}}{\int_{0}^{h_{p}}{I_{0}e^{j\; \Phi}{\cos \left( {\beta_{0}z} \right)}{dz}}}} \cong {h_{p}e^{j\; \Phi}}}},} & (37)\end{matrix}$

for the case where the physical height h_(p)<<λ₀. The complex effectiveheight of a monopole, h_(eff)=h_(p) at an angle (or phase delay) of Φ,may be adjusted to cause the source fields to match a guided surfacewaveguide mode and cause a guided surface wave to be launched on thelossy conducting medium 203.

In the example of FIG. 5A, ray optics are used to illustrate the complexangle trigonometry of the incident electric field (E) having a complexBrewster angle of incidence (θ_(i,B)) at the Hankel crossover distance(R_(x)) 121. Recall from Equation (26) that, for a lossy conductingmedium, the Brewster angle is complex and specified by

$\begin{matrix}{{\tan \; \theta_{i,B}} = {\sqrt{ɛ_{r} - {j\frac{\sigma}{{\omega ɛ}_{o}}}} = {n.}}} & (38)\end{matrix}$

Electrically, the geometric parameters are related by the electricaleffective height (h_(eff)) of the charge terminal T₁ by

$\begin{matrix}{{{R_{x}\tan \; \psi_{i,B}} = {{R_{x} \times W} = {h_{eff} = {h_{p}e^{j\; \Phi}}}}},} & (39)\end{matrix}$

where ψ_(i,B)=(π/2)−θ_(i,B) is the Brewster angle measured from thesurface of the lossy conducting medium. To couple into the guidedsurface waveguide mode, the wave tilt of the electric field at theHankel crossover distance can be expressed as the ratio of theelectrical effective height and the Hankel crossover distance

$\begin{matrix}{\frac{h_{eff}}{R_{x}} = {{\tan \; \psi_{i,B}} = {W_{Rx}.}}} & (40)\end{matrix}$

Since both the physical height (h_(p)) and the Hankel crossover distance(R_(x)) are real quantities, the angle (Ψ) of the desired guided surfacewave tilt at the Hankel crossover distance (R_(x)) is equal to the phase(Φ) of the complex effective height (h_(eff)). This implies that byvarying the phase at the supply point of the coil, and thus the phasedelay in Equation (37), the phase, Φ, of the complex effective heightcan be manipulated to match the angle of the wave tilt, Ψ, of the guidedsurface waveguide mode at the Hankel crossover point 121: Φ=Ψ.

In FIG. 5A, a right triangle is depicted having an adjacent side oflength R_(x) along the lossy conducting medium surface and a complexBrewster angle ψ_(i,B), measured between a ray 124 extending between theHankel crossover point 121 at R_(x) and the center of the chargeterminal T₁, and the lossy conducting medium surface 127 between theHankel crossover point 121 and the charge terminal T₁. With the chargeterminal T₁ positioned at physical height h_(p) and excited with acharge having the appropriate phase delay Φ, the resulting electricfield is incident with the lossy conducting medium boundary interface atthe Hankel crossover distance R_(x), and at the Brewster angle. Underthese conditions, the guided surface waveguide mode can be excitedwithout reflection or substantially negligible reflection.

If the physical height of the charge terminal T₁ is decreased withoutchanging the phase delay t of the effective height (h_(eff)), theresulting electric field intersects the lossy conducting medium 203 atthe Brewster angle at a reduced distance from the guided surfacewaveguide probe 200. FIG. 6 graphically illustrates the effect ofdecreasing the physical height of the charge terminal T₁ on the distancewhere the electric field is incident at the Brewster angle. As theheight is decreased from h₃ through h₂ to h₁, the point where theelectric field intersects with the lossy conducting medium (e.g., theEarth) at the Brewster angle moves closer to the charge terminalposition. However, as Equation (39) indicates, the height H₁ (FIG. 3) ofthe charge terminal T₁ should be at or higher than the physical height(h_(p)) in order to excite the far-out component of the Hankel function.With the charge terminal T₁ positioned at or above the effective height(h_(eff)), the lossy conducting medium 203 can be illuminated at theBrewster angle of incidence (ψ_(i,B)=(π/2)−θ_(i,B)) at or beyond theHankel crossover distance (R_(x)) 121 as illustrated in FIG. 5A. Toreduce or minimize the bound charge on the charge terminal T₁, theheight should be at least four times the spherical diameter (orequivalent spherical diameter) of the charge terminal T₁ as mentionedabove.

A guided surface waveguide probe 200 can be configured to establish anelectric field having a wave tilt that corresponds to a waveilluminating the surface of the lossy conducting medium 203 at a complexBrewster angle, thereby exciting radial surface currents bysubstantially mode-matching to a guided surface wave mode at (or beyond)the Hankel crossover point 121 at R_(x).

Referring to FIG. 7A, shown is a graphical representation of an exampleof a guided surface waveguide probe 200 b that includes a chargeterminal T₁. As shown in FIG. 7A, an excitation source 212 such as an ACsource acts as the excitation source for the charge terminal T₁, whichis coupled to the guided surface waveguide probe 200 b through a feednetwork 209 (FIG. 3) comprising a coil 215 such as, e.g., a helicalcoil. In other implementations, the excitation source 212 can beinductively coupled to the coil 215 through a primary coil. In someembodiments, an impedance matching network may be included to improveand/or maximize coupling of the excitation source 212 to the coil 215.

As shown in FIG. 7A, the guided surface waveguide probe 200 b caninclude the upper charge terminal T₁ (e.g., a sphere at height h_(p))that is positioned along a vertical axis z that is substantially normalto the plane presented by the lossy conducting medium 203. A secondmedium 206 is located above the lossy conducting medium 203. The chargeterminal T₁ has a self-capacitance C_(T). During operation, charge Q₁ isimposed on the terminal T₁ depending on the voltage applied to theterminal T₁ at any given instant.

In the example of FIG. 7A, the coil 215 is coupled to a ground stake (orgrounding system) 218 at a first end and to the charge terminal T₁ via avertical feed line conductor 221. In some implementations, the coilconnection to the charge terminal T₁ can be adjusted using a tap 224 ofthe coil 215 as shown in FIG. 7A. The coil 215 can be energized at anoperating frequency by the excitation source 212 comprising, forexample, an excitation source through a tap 227 at a lower portion ofthe coil 215. In other implementations, the excitation source 212 can beinductively coupled to the coil 215 through a primary coil. The chargeterminal T₁ can be configured to adjust its load impedance seen by thevertical feed line conductor 221, which can be used to adjust the probeimpedance.

FIG. 7B shows a graphical representation of another example of a guidedsurface waveguide probe 200 c that includes a charge terminal T₁. As inFIG. 7A, the guided surface waveguide probe 200 c can include the uppercharge terminal T₁ positioned over the lossy conducting medium 203(e.g., at height h_(p)). In the example of FIG. 7B, the phasing coil 215is coupled at a first end to a ground stake (or grounding system) 218via a lumped element tank circuit 260 and to the charge terminal T₁ at asecond end via a vertical feed line conductor 221. The phasing coil 215can be energized at an operating frequency by the excitation source 212through, e.g., a tap 227 at a lower portion of the coil 215, as shown inFIG. 7A. In other implementations, the excitation source 212 can beinductively coupled to the phasing coil 215 or an inductive coil 263 ofa tank circuit 260 through a primary coil 269. The inductive coil 263may also be called a “lumped element” coil as it behaves as a lumpedelement or inductor. In the example of FIG. 7B, the phasing coil 215 isenergized by the excitation source 212 through inductive coupling withthe inductive coil 263 of the lumped element tank circuit 260. Thelumped element tank circuit 260 comprises the inductive coil 263 and acapacitor 266. The inductive coil 263 and/or the capacitor 266 can befixed or variable to allow for adjustment of the tank circuit resonance,and thus the probe impedance.

FIG. 7C shows a graphical representation of another example of a guidedsurface waveguide probe 200 d that includes a charge terminal T₁. As inFIG. 7A, the guided surface waveguide probe 200 d can include the uppercharge terminal T₁ positioned over the lossy conducting medium 203(e.g., at height h_(p)). The feed network 209 can comprise a pluralityof phasing coils (e.g., helical coils) instead of a single phasing coil215 as illustrated in FIGS. 7A and 7B. The plurality of phasing coilscan include a combination of helical coils to provide the appropriatephase delay (e.g., θ_(c)=θ_(ca)+θ_(cb), where θ_(ca) and θ_(cb)correspond to the phase delays of coils 215 a and 215 b, respectively)to launch a guided surface wave. In the example of FIG. 7C, the feednetwork includes two phasing coils 215 a and 215 b connected in serieswith the lower coil 215 b coupled to a ground stake (or groundingsystem) 218 via a lumped element tank circuit 260 and the upper coil 215a coupled to the charge terminal T₁ via a vertical feed line conductor221. The phasing coils 215 a and 215 b can be energized at an operatingfrequency by the excitation source 212 through, e.g., inductive couplingvia a primary coil 269 with, e.g., the upper phasing coil 215 a, thelower phasing coil 215 b, and/or an inductive coil 263 of the tankcircuit 260. For example, as shown in FIG. 7C, the coil 215 can beenergized by the excitation source 212 through inductive coupling fromthe primary coil 269 to the lower phasing coil 215 b. Alternatively, asin the example shown in FIG. 7B, the coil 215 can be energized by theexcitation source 212 through inductive coupling from the primary coil269 to the inductive coil 263 of the lumped element tank circuit 260.The inductive coil 263 and/or the capacitor 266 of the lumped elementtank circuit 260 can be fixed or variable to allow for adjustment of thetank circuit resonance, and thus the probe impedance.

At this point, it should be pointed out that there is a distinctionbetween phase delays for traveling waves and phase shifts for standingwaves. Phase delays for traveling waves, θ=βl, are due to propagationtime delays on distributed element wave guiding structures such as,e.g., the coil(s) 215 and vertical feed line conductor 221. A phasedelay is not experienced as the traveling wave passes through the lumpedelement tank circuit 260. As a result, the total traveling wave phasedelay through, e.g., the guided surface waveguide probes 200 c and 200 dis still Φ=θ_(c)+θ_(y).

However, the position dependent phase shifts of standing waves, whichcomprise forward and backward propagating waves, and load dependentphase shifts depend on both the line-length propagation delay and attransitions between line sections of different characteristicimpedances. It should be noted that phase shifts do occur in lumpedelement circuits. Phase shifts also occur at the impedancediscontinuities between transmission line segments and between linesegments and loads. This comes from the complex reflection coefficient,Γ=|Γ|e^(jϕ), arising from the impedance discontinuities, and results instanding waves (wave interference patterns of forward and backwardpropagating waves) on the distributed element structures. As a result,the total standing wave phase shift of the guided surface waveguideprobes 200 c and 200 d includes the phase shift produced by the lumpedelement tank circuit 260.

Accordingly, it should be noted that coils that produce both a phasedelay for a traveling wave and a phase shift for standing waves can bereferred to herein as “phasing coils.” The coils 215 are examples ofphasing coils. It should be further noted that coils in a tank circuit,such as the lumped element tank circuit 260 as described above, act as alumped element and an inductor, where the tank circuit produces a phaseshift for standing waves without a corresponding phase delay fortraveling waves. Such coils acting as lumped elements or inductors canbe referred to herein as “inductor coils” or “lumped element” coils.Inductive coil 263 is an example of such an inductor coil or lumpedelement coil. Such inductor coils or lumped element coils are assumed tohave a uniform current distribution throughout the coil, and areelectrically small relative to the wavelength of operation of the guidedsurface waveguide probe 200 such that they produce a negligible delay ofa traveling wave.

The construction and adjustment of the guided surface waveguide probe200 is based upon various operating conditions, such as the transmissionfrequency, conditions of the lossy conducting medium (e.g., soilconductivity σ and relative permittivity ε_(r)), and size of the chargeterminal T₁. The index of refraction can be calculated from Equations(10) and (11) as

n=√{square root over (ε_(r) −jx)},  (41)

where x=σ/ωε_(o) with ω=2πf. The conductivity σ and relativepermittivity ε_(r) can be determined through test measurements of thelossy conducting medium 203. The complex Brewster angle (θ_(i,B))measured from the surface normal can also be determined from Equation(26) as

θ_(i,B)=arctan(√{square root over (ε_(r) −jx)}),  (42)

or measured from the surface as shown in FIG. 5A as

$\begin{matrix}{\psi_{i,B} = {\frac{\pi}{2} - {\theta_{i,B}.}}} & (43)\end{matrix}$

The wave tilt at the Hankel crossover distance (W_(Rx)) can also befound using Equation (40).

The Hankel crossover distance can also be found by equating themagnitudes of Equations (20b) and (21) for −jγρ, and solving for R_(x)as illustrated by FIG. 4. The electrical effective height can then bedetermined from Equation (39) using the Hankel crossover distance andthe complex Brewster angle as

$\begin{matrix}{h_{eff} = {{h_{p}e^{j\; \Phi}} = {R_{x}\tan \; {\psi_{i,B}.}}}} & (44)\end{matrix}$

As can be seen from Equation (44), the complex effective height(h_(eff)) includes a magnitude that is associated with the physicalheight (h_(p)) of the charge terminal T₁ and a phase delay (Φ) that isto be associated with the angle (Ψ) of the wave tilt at the Hankelcrossover distance (R_(x)). With these variables and the selected chargeterminal T₁ configuration, it is possible to determine the configurationof a guided surface waveguide probe 200.

With the charge terminal T₁ positioned at or above the physical height(h_(p)), the feed network 209 (FIG. 3) and/or the vertical feed lineconnecting the feed network to the charge terminal T₁ can be adjusted tomatch the phase delay (Φ) of the charge Q₁ on the charge terminal T₁ tothe angle (Ψ) of the wave tilt (W). The size of the charge terminal T₁can be chosen to provide a sufficiently large surface for the charge Q₁imposed on the terminals. In general, it is desirable to make the chargeterminal T₁ as large as practical. The size of the charge terminal T₁should be large enough to avoid ionization of the surrounding air, whichcan result in electrical discharge or sparking around the chargeterminal.

The phase delay θ_(c) of a helically-wound coil can be determined fromMaxwell's equations as has been discussed by Corum, K. L. and J. F.Corum, “RF Coils, Helical Resonators and Voltage Magnification byCoherent Spatial Modes,” Microwave Review, Vol. 7, No. 2, September2001, pp. 36-45, which is incorporated herein by reference in itsentirety. For a helical coil with H/D>1, the ratio of the velocity ofpropagation (v) of a wave along the coil's longitudinal axis to thespeed of light (c), or the “velocity factor,” is given by

$\begin{matrix}{{V_{f} = {\frac{\upsilon}{c} = \frac{1}{\sqrt{1 + {20\left( \frac{D}{s} \right)^{2.5}\left( \frac{D}{\lambda_{0}} \right)^{0.5}}}}}},} & (45)\end{matrix}$

where H is the axial length of the solenoidal helix, D is the coildiameter, N is the number of turns of the coil, s=H/N is theturn-to-turn spacing (or helix pitch) of the coil, and λ₀ is thefree-space wavelength. Based upon this relationship, the electricallength, or phase delay, of the helical coil is given by

$\begin{matrix}{\theta_{c} = {{\beta_{p}H} = {{\frac{2\pi}{\lambda_{p}}H} = {\frac{2\pi}{V_{f}\lambda_{0}}{H.}}}}} & (46)\end{matrix}$

The principle is the same if the helix is wound spirally or is short andfat, but V_(f) and θ_(c) are easier to obtain by experimentalmeasurement. The expression for the characteristic (wave) impedance of ahelical transmission line has also been derived as

$\begin{matrix}{Z_{c} = {{\frac{60}{V_{f}}\left\lbrack {{\; {n\left( \frac{V_{f}\lambda_{0}}{D} \right)}} - 1.027} \right\rbrack}.}} & (47)\end{matrix}$

The spatial phase delay θ_(y) of the structure can be determined usingthe traveling wave phase delay of the vertical feed line conductor 221(FIGS. 7A-7C). The capacitance of a cylindrical vertical conductor abovea prefect ground plane can be expressed as

$\begin{matrix}{{C_{A} = {\frac{2{\pi ɛ}_{o}h_{w}}{{\; {n\left( \frac{h}{a} \right)}} - 1}\mspace{14mu} {Farads}}},} & (48)\end{matrix}$

where h_(w) is the vertical length (or height) of the conductor and a isthe radius (in mks units). As with the helical coil, the traveling wavephase delay of the vertical feed line conductor can be given by

$\begin{matrix}{{\theta_{y} = {{\beta_{w}h_{w}} = {{\frac{2\pi}{\lambda_{w}}h_{w}} = {\frac{2\pi}{V_{w}\lambda_{0}}h_{w}}}}},} & (49)\end{matrix}$

where β_(w) is the propagation phase constant for the vertical feed lineconductor, h_(w) is the vertical length (or height) of the vertical feedline conductor, V_(w) is the velocity factor on the wire, λ₀ is thewavelength at the supplied frequency, and λ_(w) is the propagationwavelength resulting from the velocity factor V_(w). For a uniformcylindrical conductor, the velocity factor is a constant withV_(w)≈0.94, or in a range from about 0.93 to about 0.98. If the mast isconsidered to be a uniform transmission line, its average characteristicimpedance can be approximated by

$\begin{matrix}{{Z_{w} = {\frac{60}{V_{w}}\left\lbrack {{\; {n\left( \frac{h_{w}}{a} \right)}} - 1} \right\rbrack}},} & (50)\end{matrix}$

where V_(w)≈0.94 for a uniform cylindrical conductor and a is the radiusof the conductor. An alternative expression that has been employed inamateur radio literature for the characteristic impedance of asingle-wire feed line can be given by

$\begin{matrix}{Z_{w} = {138\; {{\log \left( \frac{1.123\; V_{w}\lambda_{0}}{2\pi \; a} \right)}.}}} & (51)\end{matrix}$

Equation (51) implies that Z_(w) for a single-wire feeder varies withfrequency. The phase delay can be determined based upon the capacitanceand characteristic impedance.

With a charge terminal T₁ positioned over the lossy conducting medium203 as shown in FIG. 3, the feed network 209 can be adjusted to excitethe charge terminal T₁ with the phase delay (Φ) of the complex effectiveheight (h_(eff)) equal to the angle (Ψ) of the wave tilt at the Hankelcrossover distance, or Φ=Ψ. When this condition is met, the electricfield produced by the charge oscillating Q₁ on the charge terminal T₁ iscoupled into a guided surface waveguide mode traveling along the surfaceof a lossy conducting medium 203. For example, if the Brewster angle(θ_(i,B)), the phase delay (θ_(y)) associated with the vertical feedline conductor 221 (FIGS. 7A-7C), and the configuration of the coil(s)215 (FIGS. 7A-7C) are known, then the position of the tap 224 (FIGS.7A-7C) can be determined and adjusted to impose an oscillating charge Q₁on the charge terminal T₁ with phase Φ=Ψ. The position of the tap 224may be adjusted to maximize coupling the traveling surface waves intothe guided surface waveguide mode. Excess coil length beyond theposition of the tap 224 can be removed to reduce the capacitive effects.The vertical wire height and/or the geometrical parameters of thehelical coil may also be varied.

The coupling to the guided surface waveguide mode on the surface of thelossy conducting medium 203 can be improved and/or optimized by tuningthe guided surface waveguide probe 200 for standing wave resonance withrespect to a complex image plane associated with the charge Q₁ on thecharge terminal T₁. By doing this, the performance of the guided surfacewaveguide probe 200 can be adjusted for increased and/or maximum voltage(and thus charge Q₁) on the charge terminal T₁. Referring back to FIG.3, the effect of the lossy conducting medium 203 in Region 1 can beexamined using image theory analysis.

Physically, an elevated charge Q₁ placed over a perfectly conductingplane attracts the free charge on the perfectly conducting plane, whichthen “piles up” in the region under the elevated charge Q₁. Theresulting distribution of “bound” electricity on the perfectlyconducting plane is similar to a bell-shaped curve. The superposition ofthe potential of the elevated charge Q₁, plus the potential of theinduced “piled up” charge beneath it, forces a zero equipotentialsurface for the perfectly conducting plane. The boundary value problemsolution that describes the fields in the region above the perfectlyconducting plane may be obtained using the classical notion of imagecharges, where the field from the elevated charge is superimposed withthe field from a corresponding “image” charge below the perfectlyconducting plane.

This analysis may also be used with respect to a lossy conducting medium203 by assuming the presence of an effective image charge Q₁′ beneaththe guided surface waveguide probe 200. The effective image charge Q₁′coincides with the charge Q₁ on the charge terminal T₁ about aconducting image ground plane 130, as illustrated in FIG. 3. However,the image charge Q₁′ is not merely located at some real depth and 180°out of phase with the primary source charge Q₁ on the charge terminalT₁, as they would be in the case of a perfect conductor. Rather, thelossy conducting medium 203 (e.g., a terrestrial medium) presents aphase shifted image. That is to say, the image charge Q₁′ is at acomplex depth below the surface (or physical boundary) of the lossyconducting medium 203. For a discussion of complex image depth,reference is made to Wait, J. R., “Complex Image Theory—Revisited,” IEEEAntennas and Propagation Magazine, Vol. 33, No. 4, August 1991, pp.27-29, which is incorporated herein by reference in its entirety.

Instead of the image charge Q₁′ being at a depth that is equal to thephysical height (H₁) of the charge Q₁, the conducting image ground plane130 (representing a perfect conductor) is located at a complex depth ofz=−d/2 and the image charge Q₁′ appears at a complex depth (i.e., the“depth” has both magnitude and phase), given by −D₁=−(d/2+d/2+H₁)≠H₁.For vertically polarized sources over the Earth,

$\begin{matrix}{{d = {{\frac{\sqrt[2]{\gamma_{e}^{2} + k_{0}^{2}}}{\gamma_{e}^{2}} \approx \frac{2}{\gamma_{e}}} = {{d_{r} + {jd}_{i}} = {{d}{\angle\zeta}}}}},} & (52)\end{matrix}$where

γ_(e) ² =jωμ ₁σ₁−ω²μ₁ϵ₁, and  (53)

k _(o)=ω√{square root over (μ_(o)ε_(o))},  (54)

as indicated in Equation (12). The complex spacing of the image charge,in turn, implies that the external field will experience extra phaseshifts not encountered when the interface is either a dielectric or aperfect conductor. In the lossy conducting medium, the wave front normalis parallel to the tangent of the conducting image ground plane 130 atz=−d/2, and not at the boundary interface between Regions 1 and 2.

Consider the case illustrated in FIG. 8A where the lossy conductingmedium 203 is a finitely conducting Earth 133 with a physical boundary136. The finitely conducting Earth 133 may be replaced by a perfectlyconducting image ground plane 139 as shown in FIG. 8B, which is locatedat a complex depth z₁ below the physical boundary 136. This equivalentrepresentation exhibits the same impedance when looking down into theinterface at the physical boundary 136. The equivalent representation ofFIG. 8B can be modeled as an equivalent transmission line, as shown inFIG. 8C. The cross-section of the equivalent structure is represented asa (z-directed) end-loaded transmission line, with the impedance of theperfectly conducting image plane being a short circuit (z_(s)=0). Thedepth z₁ can be determined by equating the TEM wave impedance lookingdown at the Earth to an image ground plane impedance z_(in) seen lookinginto the transmission line of FIG. 8C.

In the case of FIG. 8A, the propagation constant and wave intrinsicimpedance in the upper region (air) 142 are

$\begin{matrix}{{\gamma_{o} = {{j\; \omega \sqrt{\mu_{o}ɛ_{o}}} = {0 + {j\; \beta_{o}}}}},{and}} & (55) \\{z_{o} = {\frac{j\; {\omega\mu}_{o}}{\gamma_{o}} = {\sqrt{\frac{\mu_{o}}{ɛ_{o}}}.}}} & (56)\end{matrix}$

In the lossy Earth 133, the propagation constant and wave intrinsicimpedance are

$\begin{matrix}{{\gamma_{e} = \sqrt{j\; {{\omega\mu}_{1}\left( {\sigma_{1} + {j\; {\omega ɛ}_{1}}} \right)}}},{and}} & (57) \\{Z_{e} = {\frac{j\; {\omega\mu}_{1}}{\gamma_{e}}.}} & (58)\end{matrix}$

For normal incidence, the equivalent representation of FIG. 8B isequivalent to a TEM transmission line whose characteristic impedance isthat of air (z_(o)), with propagation constant of γ_(o), and whoselength is z₁. As such, the image ground plane impedance Z_(in) seen atthe interface for the shorted transmission line of FIG. 8C is given by

Z _(in) =Z _(o) tan h(γ_(o) z ₁).  (59)

Equating the image ground plane impedance Z_(in) associated with theequivalent model of FIG. 8C to the normal incidence wave impedance ofFIG. 8A and solving for z₁ gives the distance to a short circuit (theperfectly conducting image ground plane 139) as

$\begin{matrix}{{z_{1} = {{\frac{1}{\gamma_{o}}{\tanh^{- 1}\left( \frac{Z_{e}}{Z_{o}} \right)}} = {{\frac{1}{\gamma_{o}}{\tanh^{- 1}\left( \frac{\gamma_{o}}{\gamma_{e}} \right)}} \approx \frac{1}{\gamma_{e}}}}},} & (60)\end{matrix}$

where only the first term of the series expansion for the inversehyperbolic tangent is considered for this approximation. Note that inthe air region 142, the propagation constant is γ_(o)=Jβ_(o), soZ_(in)=jZ_(o) tan β_(o)z₁ (which is a purely imaginary quantity for areal z₁), but z_(e) is a complex value if σ≠0. Therefore, Z_(in)=Z_(e)only when z₁ is a complex distance.

Since the equivalent representation of FIG. 8B includes a perfectlyconducting image ground plane 139, the image depth for a charge orcurrent lying at the surface of the Earth (physical boundary 136) isequal to distance z₁ on the other side of the image ground plane 139, ord=2×z₁ beneath the Earth's surface (which is located at z=0). Thus, thedistance to the perfectly conducting image ground plane 139 can beapproximated by

$\begin{matrix}{d = {{2\; z_{1}} \approx {\frac{2}{\gamma_{e}}.}}} & (61)\end{matrix}$

Additionally, the “image charge” will be “equal and opposite” to thereal charge, so the potential of the perfectly conducting image groundplane 139 at depth z₁=−d/2 will be zero.

If a charge Q₁ is elevated a distance H₁ above the surface of the Earthas illustrated in FIG. 3, then the image charge Q₁′ resides at a complexdistance of D₁=d+H₁ below the surface, or a complex distance of d/2+H₁below the image ground plane 130. The guided surface waveguide probes200 of FIGS. 7A-7C can be modeled as an equivalent single-wiretransmission line image plane model that can be based upon the perfectlyconducting image ground plane 139 of FIG. 8B.

FIG. 9A shows an example of the equivalent single-wire transmission lineimage plane model, and FIG. 9B illustrates an example of the equivalentclassic transmission line model, including the shorted transmission lineof FIG. 8C. FIG. 9C illustrates an example of the equivalent classictransmission line model including the lumped element tank circuit 260.

In the equivalent image plane models of FIGS. 9A-9C, Φ=θ_(y)+θ_(c) isthe traveling wave phase delay of the guided surface waveguide probe 200referenced to Earth 133 (or the lossy conducting medium 203),θ_(c)=β_(p)H is the electrical length of the coil or coils 215 (FIGS.7A-7C), of physical length H, expressed in degrees, θ_(y)=β_(w)h_(w) isthe electrical length of the vertical feed line conductor 221 (FIGS.7A-7C), of physical length h_(W), expressed in degrees. In addition,θ_(d)=β_(o) d/2 is the phase shift between the image ground plane 139and the physical boundary 136 of the Earth 133 (or lossy conductingmedium 203). In the example of FIGS. 9A-9C, Z_(w) is the characteristicimpedance of the elevated vertical feed line conductor 221 in ohms,Z_(c), is the characteristic impedance of the coil(s) 215 in ohms, andZ_(O) is the characteristic impedance of free space. In the example ofFIG. 9C, Z_(t) is the characteristic impedance of the lumped elementtank circuit 260 in ohms and θ_(t) is the corresponding phase shift atthe operating frequency.

At the base of the guided surface waveguide probe 200, the impedanceseen “looking up” into the structure is Z_(↑)=Z_(base). With a loadimpedance of:

$\begin{matrix}{{Z_{L} = \frac{1}{j\; \omega \; C_{T}}},} & (62)\end{matrix}$

where C_(T) is the self-capacitance of the charge terminal T₁, theimpedance seen “looking up” into the vertical feed line conductor 221(FIGS. 7A-7C) is given by:

$\begin{matrix}{{Z_{2} = {{Z_{W}\frac{Z_{L} + {Z_{w}{\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}{Z_{w} + {Z_{L}{\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}} = {Z_{W}\frac{Z_{L} + {Z_{w}{\tanh \left( {j\; \theta_{y}} \right)}}}{Z_{W} + {Z_{L}{\tanh \left( {j\; \theta_{y}} \right)}}}}}},} & (63)\end{matrix}$

and the impedance seen “looking up” into the coil 215 (FIGS. 7A and 7B)is given by:

$\begin{matrix}{Z_{base} = {{Z_{c}\frac{Z_{2} + {Z_{c}{\tanh \left( {j\; \beta_{p}H} \right)}}}{Z_{c} + {Z_{2}{\tanh \left( {j\; \beta_{p}H} \right)}}}} = {Z_{c}{\frac{Z_{2} + {Z_{c}{\tanh \left( {j\; \theta_{c}} \right)}}}{Z_{c} + {Z_{2}{\tanh \left( {j\; \theta_{c}} \right)}}}.}}}} & (64)\end{matrix}$

Where the feed network 209 includes a plurality of coils 215 (e.g., FIG.7C), the impedance seen at the base of each coil 215 can be sequentiallydetermined using Equation (64). For example, the impedance seen “lookingup” into the upper coil 215 a of FIG. 7C is given by:

$\begin{matrix}{{Z_{coil} = {{Z_{ca}\frac{Z_{2} + {Z_{ca}{\tanh \left( {j\; \beta_{p}H} \right)}}}{Z_{ca} + {Z_{2}{\tanh \left( {j\; \beta_{p}H} \right)}}}} = {Z_{ca}\frac{Z_{2} + {Z_{ca}{\tanh \left( {j\; \theta_{ca}} \right)}}}{Z_{ca} + {Z_{2}{\tanh \left( {j\; \theta_{ca}} \right)}}}}}},} & (64.1)\end{matrix}$

and the impedance seen “looking up” into the lower coil 215 b of FIG. 7Ccan be given by:

$\begin{matrix}{{Z_{base} = {{Z_{cb}\frac{Z_{coil} + {Z_{cb}{\tanh \left( {j\; \beta_{p}H} \right)}}}{Z_{cb} + {Z_{coil}{\tanh \left( {j\; \beta_{p}H} \right)}}}} = {Z_{cb}\frac{Z_{coil} + {Z_{cb}{\tanh \left( {j\; \theta_{cb}} \right)}}}{Z_{cb} + {Z_{coil}{\tanh \left( {j\; \theta_{c}b} \right)}}}}}},} & (64.2)\end{matrix}$

where Z_(ca) and Z_(cb) are the characteristic impedances of the upperand lower coils. This can be extended to account for additional coils215 as needed. At the base of the guided surface waveguide probe 200,the impedance seen “looking down” into the lossy conducting medium 203is Z_(↓)=Z_(in), which is given by:

$\begin{matrix}{{Z_{in} = {{Z_{o}\frac{Z_{s} + {Z_{o}{\tanh \left\lbrack {j\; {\beta_{o}\left( {d/2} \right)}} \right\rbrack}}}{Z_{o} + {Z_{s}{\tanh \left\lbrack {j\; {\beta_{o}\left( {d/2} \right)}} \right\rbrack}}}} = {Z_{o}{\tanh \left( {j\; \theta_{d}} \right)}}}},} & (65)\end{matrix}$

where Z_(s)=0.

Neglecting losses, the equivalent image plane model can be tuned toresonance when Z_(↓)+Z_(↑)=0 at the physical boundary 136. Or, in thelow loss case, X_(↓)+X_(↑)=0 at the physical boundary 136, where X isthe corresponding reactive component. Thus, the impedance at thephysical boundary 136 “looking up” into the guided surface waveguideprobe 200 is the conjugate of the impedance at the physical boundary 136“looking down” into the lossy conducting medium 203. By adjusting theprobe impedance via the load impedance Z_(L) of the charge terminal T₁while maintaining the traveling wave phase delay Φ equal to the angle ofthe media's wave tilt Ψ, so that Φ=Ψ, which improves and/or maximizescoupling of the probe's electric field to a guided surface waveguidemode along the surface of the lossy conducting medium 203 (e.g., Earth),the equivalent image plane models of FIGS. 9A and 9B can be tuned toresonance with respect to the image ground plane 139. In this way, theimpedance of the equivalent complex image plane model is purelyresistive, which maintains a superposed standing wave on the probestructure that maximizes the voltage and elevated charge on terminal T₁,and by equations (1)-(3) and (16) maximizes the propagating surfacewave.

While the load impedance Z_(L) of the charge terminal T₁ can be adjustedto tune the probe 200 for standing wave resonance with respect to theimage ground plane 139, in some embodiments a lumped element tankcircuit 260 located between the coil(s) 215 (FIGS. 7B and 7C) and theground stake (or grounding system) 218 can be adjusted to tune the probe200 for standing wave resonance with respect to the image ground plane139 as illustrated in FIG. 9C. A phase delay is not experienced as thetraveling wave passes through the lumped element tank circuit 260. As aresult, the total traveling wave phase delay through, e.g., the guidedsurface waveguide probes 200 c and 200 d is still ψ=θ_(c)+θ_(y).However, it should be noted that phase shifts do occur in lumped elementcircuits. Phase shifts also occur at impedance discontinuities betweentransmission line segments and between line segments and loads. Thus,the tank circuit 260 may also be referred to as a “phase shift circuit.”

With the lumped element tank circuit 260 coupled to the base of theguided surface waveguide probe 200, the impedance seen “looking up” intothe tank circuit 260 is Z_(↑)=Z_(tuning), which can be given by:

Z _(tuning) =Z _(base) −Z _(t),

where Z_(t) is the characteristic impedance of the tank circuit 260 andZ_(base) is the impedance seen “looking up” into the coil(s) as givenin, e.g., Equations (64) or (64.2). FIG. 9D illustrates the variation ofthe impedance of the lumped element tank circuit 260 with respect tooperating frequency (f_(o)) based upon the resonant frequency (f_(p)) ofthe tank circuit 260. As shown in FIG. 9D, the impedance of the lumpedelement tank 260 can be inductive or capacitive depending on the tunedself-resonant frequency of the tank circuit. When operating the tankcircuit 260 at a frequency below its self-resonant frequency (f_(p)),its terminal point impedance is inductive, and for operation above f_(p)the terminal point impedance is capacitive. Adjusting either theinductance 263 or the capacitance 266 of the tank circuit 260 changesf_(p) and shifts the impedance curve in FIG. 9D, which affects theterminal point impedance seen at a given operating frequency f_(o).

Neglecting losses, the equivalent image plane model with the tankcircuit 260 can be tuned to resonance when Z_(↓)+Z_(↑)=0 at the physicalboundary 136. Or, in the low loss case, X_(↓)+X_(↑)=0 at the physicalboundary 136, where X is the corresponding reactive component. Thus, theimpedance at the physical boundary 136 “looking up” into the lumpedelement tank circuit 260 is the conjugate of the impedance at thephysical boundary 136 “looking down” into the lossy conducting medium203. By adjusting the lumped element tank circuit 260 while maintainingthe traveling wave phase delay Φ equal to the angle of the media's wavetilt Ψ, so that Φ=Ψ, the equivalent image plane models can be tuned toresonance with respect to the image ground plane 139. In this way, theimpedance of the equivalent complex image plane model is purelyresistive, which maintains a superposed standing wave on the probestructure that maximizes the voltage and elevated charge on terminal T₁,and improves and/or maximizes coupling of the probe's electric field toa guided surface waveguide mode along the surface of the lossyconducting medium 203 (e.g., earth).

It follows from the Hankel solutions, that the guided surface waveexcited by the guided surface waveguide probe 200 is an outwardpropagating traveling wave. The source distribution along the feednetwork 209 between the charge terminal T₁ and the ground stake (orgrounding system) 218 of the guided surface waveguide probe 200 (FIGS. 3and 7A-7C) is actually composed of a superposition of a traveling waveplus a standing wave on the structure. With the charge terminal T₁positioned at or above the physical height h_(p), the phase delay of thetraveling wave moving through the feed network 209 is matched to theangle of the wave tilt associated with the lossy conducting medium 203.This mode-matching allows the traveling wave to be launched along thelossy conducting medium 203. Once the phase delay has been establishedfor the traveling wave, the load impedance Z_(L) of the charge terminalT₁ and/or the lumped element tank circuit 260 can be adjusted to bringthe probe structure into standing wave resonance with respect to theimage ground plane (130 of FIG. 3 or 139 of FIG. 8), which is at acomplex depth of −d/2. In that case, the impedance seen from the imageground plane has zero reactance and the charge on the charge terminal T₁is maximized.

The distinction between the traveling wave phenomenon and standing wavephenomena is that (1) the phase delay of traveling waves (θ=βd) on asection of transmission line of length d (sometimes called a “delayline”) is due to propagation time delays; whereas (2) theposition-dependent phase of standing waves (which are composed offorward and backward propagating waves) depends on both the line lengthpropagation time delay and impedance transitions at interfaces betweenline sections of different characteristic impedances. In addition to thephase delay that arises due to the physical length of a section oftransmission line operating in sinusoidal steady-state, there is anextra reflection coefficient phase at impedance discontinuities that isdue to the ratio of Z_(oa)/Z_(ob), where Z_(oa) and Z_(ob) are thecharacteristic impedances of two sections of a transmission line suchas, e.g., a helical coil section of characteristic impedanceZ_(oa)=Z_(c) (FIG. 9B) and a straight section of vertical feed lineconductor of characteristic impedance Z_(ob)=Z_(w) (FIG. 9B).

As a result of this phenomenon, two relatively short transmission linesections of widely differing characteristic impedance may be used toprovide a very large phase shift. For example, a probe structurecomposed of two sections of transmission line, one of low impedance andone of high impedance, together totaling a physical length of, say,0.05λ, may be fabricated to provide a phase shift of 90°, which isequivalent to a 0.25λ resonance. This is due to the large jump incharacteristic impedances. In this way, a physically short probestructure can be electrically longer than the two physical lengthscombined. This is illustrated in FIGS. 9A and 9B, where thediscontinuities in the impedance ratios provide large jumps in phase.The impedance discontinuity provides a substantial phase shift where thesections are joined together.

Referring to FIG. 10, shown is a flow chart 150 illustrating an exampleof adjusting a guided surface waveguide probe 200 (FIGS. 3 and 7A-7C) tosubstantially mode-match to a guided surface waveguide mode on thesurface of the lossy conducting medium, which launches a guided surfacetraveling wave along the surface of a lossy conducting medium 203 (FIGS.3 and 7A-7C). Beginning with 153, the charge terminal T₁ of the guidedsurface waveguide probe 200 is positioned at a defined height above alossy conducting medium 203. Utilizing the characteristics of the lossyconducting medium 203 and the operating frequency of the guided surfacewaveguide probe 200, the Hankel crossover distance can also be found byequating the magnitudes of Equations (20b) and (21) for −jγρ, andsolving for R_(x) as illustrated by FIG. 4. The complex index ofrefraction (n) can be determined using Equation (41), and the complexBrewster angle (θ_(i,B)) can then be determined from Equation (42). Thephysical height (h_(p)) of the charge terminal T₁ can then be determinedfrom Equation (44). The charge terminal T₁ should be at or higher thanthe physical height (h_(p)) in order to excite the far-out component ofthe Hankel function. This height relationship is initially consideredwhen launching surface waves. To reduce or minimize the bound charge onthe charge terminal T₁, the height should be at least four times thespherical diameter (or equivalent spherical diameter) of the chargeterminal T₁.

At 156, the electrical phase delay Φ of the elevated charge Q₁ on thecharge terminal T₁ is matched to the complex wave tilt angle Ψ. Thephase delay (θ_(c)) of the helical coil(s) and/or the phase delay(θ_(y)) of the vertical feed line conductor can be adjusted to make Φequal to the angle (Ψ) of the wave tilt (W). Based on Equation (31), theangle (Ψ) of the wave tilt can be determined from:

$\begin{matrix}{W = {\frac{E_{\rho}}{E_{z}} = {\frac{1}{\tan \; \theta_{i,B}} = {\frac{1}{n} = {{W}{e^{j\; \Psi}.}}}}}} & (66)\end{matrix}$

The electrical phase delay Φ can then be matched to the angle of thewave tilt. This angular (or phase) relationship is next considered whenlaunching surface waves. For example, the electrical phase delayΦ=θ_(c)+θ_(y) can be adjusted by varying the geometrical parameters ofthe coil(s) 215 (FIGS. 7A-7C) and/or the length (or height) of thevertical feed line conductor 221 (FIGS. 7A-7C). By matching Φ=Ψ, anelectric field can be established at or beyond the Hankel crossoverdistance (R_(x)) with a complex Brewster angle at the boundary interfaceto excite the surface waveguide mode and launch a traveling wave alongthe lossy conducting medium 203.

Next at 159, the impedance of the charge terminal T₁ and/or the lumpedelement tank circuit 260 can be tuned to resonate the equivalent imageplane model of the guided surface waveguide probe 200. The depth (d/2)of the conducting image ground plane 139 of FIGS. 9A and 9B (or 130 ofFIG. 3) can be determined using Equations (52), (53) and (54) and thevalues of the lossy conducting medium 203 (e.g., the Earth), which canbe measured. Using that depth, the phase shift (θ_(d)) between the imageground plane 139 and the physical boundary 136 of the lossy conductingmedium 203 can be determined using θ_(d)=β_(o) d/2. The impedance(Z_(in)) as seen “looking down” into the lossy conducting medium 203 canthen be determined using Equation (65). This resonance relationship canbe considered to maximize the launched surface waves.

Based upon the adjusted parameters of the coil(s) 215 and the length ofthe vertical feed line conductor 221, the velocity factor, phase delay,and impedance of the coil(s) 215 and vertical feed line conductor 221can be determined using Equations (45) through (51). In addition, theself-capacitance (C_(T)) of the charge terminal T₁ can be determinedusing, e.g., Equation (24). The propagation factor (β_(p)) of thecoil(s) 215 can be determined using Equation (35) and the propagationphase constant (β_(w)) for the vertical feed line conductor 221 can bedetermined using Equation (49). Using the self-capacitance and thedetermined values of the coil(s) 215 and vertical feed line conductor221, the impedance (Z_(base)) of the guided surface waveguide probe 200as seen “looking up” into the coil(s) 215 can be determined usingEquations (62), (63), (64), (64.1) and/or (64.2).

The equivalent image plane model of the guided surface waveguide probe200 can be tuned to resonance by, e.g., adjusting the load impedanceZ_(L) such that the reactance component X_(base) of Z_(base) cancels outthe reactance component X_(in) of Z_(in), or X_(base)+X_(in)=0. Thus,the impedance at the physical boundary 136 “looking up” into the guidedsurface waveguide probe 200 is the conjugate of the impedance at thephysical boundary 136 “looking down” into the lossy conducting medium203. The load impedance Z_(L) can be adjusted by varying the capacitance(C_(T)) of the charge terminal T₁ without changing the electrical phasedelay Φ=θ_(c)+θ_(y) of the charge terminal T₁. An iterative approach maybe taken to tune the load impedance Z_(L) for resonance of theequivalent image plane model with respect to the conducting image groundplane 139 (or 130). In this way, the coupling of the electric field to aguided surface waveguide mode along the surface of the lossy conductingmedium 203 (e.g., Earth) can be improved and/or maximized.

The equivalent image plane model of the guided surface waveguide probe200 can also be tuned to resonance by, e.g., adjusting the lumpedelement tank circuit 260 such that the reactance component X_(tuning) ofZ_(tuning), cancels out the reactance component X_(in) of Z_(in), orX_(tuning)+X_(in)=0. Consider the parallel resonance curve in FIG. 9D,whose terminal point impedance at some operating frequency (f_(o)) isgiven by

${{jX}_{T}(f)} = {\frac{\left( {j\; 2\pi \; {fL}_{p}} \right)\left( {j\; 2\pi \; {fC}_{p}} \right)^{- 1}}{\left( {j\; 2\pi \; {fL}_{p}} \right) + \left( {j\; 2\pi \; {fC}_{p}} \right)^{- 1}} = {j{\frac{2\pi \; {fL}_{p}}{1 - {\left( {2\pi \; {fL}_{p}} \right)^{2}L_{p}C_{p}}}.}}}$

As C_(p) (or L_(p)) is varied, the self-resonant frequency (f_(p)) ofthe parallel tank circuit 260 changes and the terminal point reactanceX_(T)(f_(o)) at the frequency of operation varies from inductive (+) tocapacitive (−) depending on whether f_(o)<f_(p) or f_(p)<f_(o). Byadjusting f_(p), a wide range of reactance at f_(o) (e.g., a largeinductance L_(eq)(f_(o))=X_(T)(f_(o))/ω or a small capacitanceC_(eq)(f_(o))=−1/ωX_(T)(f_(o))) can be seen at the terminals of the tankcircuit 260.

To obtain the electrical phase delay (Φ) for coupling into the guidedsurface waveguide mode, the coil(s) 215 and vertical feed line conductor221 are usually less than a quarter wavelength. For this, an inductivereactance can be added by the lumped element tank circuit 260 so thatthe impedance at the physical boundary 136 “looking up” into the lumpedelement tank circuit 260 is the conjugate of the impedance at thephysical boundary 136 “looking down” into the lossy conducting medium203.

As seen in FIG. 9D, adjusting f_(p) of the tank circuit 260 (FIG. 7C)above the operating frequency (f_(o)) can provide the needed impedance,without changing the electrical phase delay Φ=θ_(c)+θ_(y) of the chargeterminal T₁, to tune for resonance of the equivalent image plane modelwith respect to the conducting image ground plane 139 (or 130). In somecases, a capacitive reactance may be needed and can be provided byadjusting f_(p) of the tank circuit 260 below the operating frequency.In this way, the coupling of the electric field to a guided surfacewaveguide mode along the surface of the lossy conducting medium 203(e.g., earth) can be improved and/or maximized.

This may be better understood by illustrating the situation with anumerical example. Consider a guided surface waveguide probe 200 b (FIG.7A) comprising a top-loaded vertical stub of physical height h_(p) witha charge terminal T₁ at the top, where the charge terminal T₁ is excitedthrough a helical coil and vertical feed line conductor at anoperational frequency (f_(o)) of 1.85 MHz. With a height (H₁) of 16 feetand the lossy conducting medium 203 (e.g., Earth) having a relativepermittivity of ε_(r)=15 and a conductivity of σ₁=0.010 mhos/m, severalsurface wave propagation parameters can be calculated for f_(o)=1.850MHz. Under these conditions, the Hankel crossover distance can be foundto be R_(x)=54.5 feet with a physical height of h_(p)=5.5 feet, which iswell below the actual height of the charge terminal T₁. While a chargeterminal height of H₁=5.5 feet could have been used, the taller probestructure reduced the bound capacitance, permitting a greater percentageof free charge on the charge terminal T₁ providing greater fieldstrength and excitation of the traveling wave.

The wave length can be determined as:

$\begin{matrix}{{\lambda_{o} = {\frac{c}{f_{o}} = {162.162\mspace{14mu} {meters}}}},} & (67)\end{matrix}$

where c is the speed of light. The complex index of refraction is:

n=√{square root over (ε_(r) −jx)}=7.529−j6.546,  (68)

from Equation (41), where x=σ₁/ωε_(o) with ω=2πf_(o), and the complexBrewster angle is:

θ_(i,B)=arctan(√{square root over (ε_(r) −jx)})=85.6−j3.744°.  (69)

from Equation (42). Using Equation (66), the wave tilt values can bedetermined to be:

$\begin{matrix}{W = {\frac{1}{\tan \; \theta_{i,B}} = {\frac{1}{n} = {{{W}e^{j\; \Psi}} = {0.101{e^{j\; 40.614{^\circ}}.}}}}}} & (70)\end{matrix}$

Thus, the helical coil can be adjusted to match Φ=Ψ=40.614°

The velocity factor of the vertical feed line conductor (approximated asa uniform cylindrical conductor with a diameter of 0.27 inches) can begiven as V_(w)≈0.93. Since h_(p)<<λ_(o), the propagation phase constantfor the vertical feed line conductor can be approximated as:

$\begin{matrix}{\beta_{w} = {\frac{2\pi}{\lambda_{w}} = {\frac{2\pi}{V_{w}\lambda_{0}} = {0.042\mspace{14mu} {m^{- 1}.}}}}} & (71)\end{matrix}$

From Equation (49) the phase delay of the vertical feed line conductoris:

θ_(y)=β_(w) h _(w)≈β_(w) h _(p)=11.640°.  (72)

By adjusting the phase delay of the helical coil so thatθ_(c)=28.9740=40.614°−11.640°, Φ will equal Ψ to match the guidedsurface waveguide mode. To illustrate the relationship between Φ and Ψ,FIG. 11 shows a plot of both over a range of frequencies. As both Φ andΨ are frequency dependent, it can be seen that their respective curvescross over each other at approximately 1.85 MHz.

For a helical coil having a conductor diameter of 0.0881 inches, a coildiameter (D) of 30 inches and a turn-to-turn spacing (s) of 4 inches,the velocity factor for the coil can be determined using Equation (45)as:

$\begin{matrix}{{V_{f} = {\frac{1}{\sqrt{1 + {20\left( \frac{D}{s} \right)^{2.5}\left( \frac{D}{\lambda_{o}} \right)^{0.5}}}} = 0.069}},} & (73)\end{matrix}$

and the propagation factor from Equation (35) is:

$\begin{matrix}{\beta_{p} = {\frac{2\pi}{V_{f}\lambda_{0}} = {0.564\mspace{14mu} {m^{- 1}.}}}} & (74)\end{matrix}$

With θ_(c)=28.974°, the axial length of the solenoidal helix (H) can bedetermined using Equation (46) such that:

$\begin{matrix}{H = {\frac{\theta_{c}}{\beta_{p}} = {35.2732\mspace{14mu} {{inches}.}}}} & (75)\end{matrix}$

This height determines the location on the helical coil where thevertical feed line conductor is connected, resulting in a coil with8.818 turns (N=H/s).

With the traveling wave phase delay of the coil and vertical feed lineconductor adjusted to match the wave tilt angle (Φ=θ_(c)+θ_(y)=Ψ), theload impedance (Z_(L)) of the charge terminal T₁ can be adjusted forstanding wave resonance of the equivalent image plane model of theguided surface waveguide probe 200. From the measured permittivity,conductivity and permeability of the Earth, the radial propagationconstant can be determined using Equation (57)

γ_(e)=√{square root over (jωu ₁(σ₁ +jωε ₁))}=0.25+j0.292 m⁻¹,  (76)

and the complex depth of the conducting image ground plane can beapproximated from Equation (52) as:

$\begin{matrix}{{{d \approx \frac{2}{\gamma_{e}}} = {3.364 + {j\; 3.963\mspace{14mu} {meters}}}},} & (77)\end{matrix}$

with a corresponding phase shift between the conducting image groundplane and the physical boundary of the Earth given by:

θ_(d)=β_(o)(d/2)=4.015−j4.73°.  (78)

Using Equation (65), the impedance seen “looking down” into the lossyconducting medium 203 (i.e., Earth) can be determined as:

Z _(in) =Z _(o) tan h(jθ _(d))=R _(in) +jX _(in)=31.191+j26.27ohms.  (79)

By matching the reactive component (X_(in)) seen “looking down” into thelossy conducting medium 203 with the reactive component (X_(base)) seen“looking up” into the guided surface waveguide probe 200, the couplinginto the guided surface waveguide mode may be maximized. This can beaccomplished by adjusting the capacitance of the charge terminal T₁without changing the traveling wave phase delays of the coil andvertical feed line conductor. For example, by adjusting the chargeterminal capacitance (C_(T)) to 61.8126 pF, the load impedance fromEquation (62) is:

$\begin{matrix}{{Z_{L} = {\frac{1}{j\; \omega \; C_{T}} = {{- j}\; 1392\mspace{14mu} {ohms}}}},} & (80)\end{matrix}$

and the reactive components at the boundary are matched.

Using Equation (51), the impedance of the vertical feed line conductor(having a diameter (2a) of 0.27 inches) is given as

$\begin{matrix}{{Z_{w} = {{138\mspace{14mu} {\log \left( \frac{1.123\; V_{w}\lambda_{0}}{2\pi \; a} \right)}} = {537.534\mspace{14mu} {ohms}}}},} & (81)\end{matrix}$

and the impedance seen “looking up” into the vertical feed lineconductor is given by Equation (63) as:

$\begin{matrix}{Z_{2} = {{Z_{W}\frac{Z_{L} + {Z_{w}{\tanh \left( {j\; \theta_{y}} \right)}}}{Z_{w} + {Z_{L}{\tanh \left( {j\; \theta_{y}} \right)}}}} = {{- j}\; 835.438\mspace{14mu} {{ohms}.}}}} & (82)\end{matrix}$

Using Equation (47), the characteristic impedance of the helical coil isgiven as

$\begin{matrix}{Z_{c}\; = {{\frac{60}{V_{f}}\left\lbrack {{\; {n\left( \frac{V_{f}\lambda_{0}}{D} \right)}} - 1.027} \right\rbrack} = {1446\mspace{14mu} {{ohms}.}}}} & (83)\end{matrix}$

and the impedance seen “looking up” into the coil at the base is givenby Equation (64) as:

$\begin{matrix}{Z_{base} = {{Z_{c}\frac{Z_{2} + {Z_{c}{\tanh \left( {j\; \theta_{c}} \right)}}}{Z_{c} + {Z_{2}{\tanh \left( {j\; \theta_{c}} \right)}}}} = {{- j}\; 26.271\mspace{14mu} {{ohms}.}}}} & (84)\end{matrix}$

When compared to the solution of Equation (79), it can be seen that thereactive components are opposite and approximately equal, and thus areconjugates of each other. Thus, the impedance (Z_(ip)) seen “looking up”into the equivalent image plane model of FIGS. 9A and 9B from theperfectly conducting image ground plane is only resistive orZ_(ip)=R+j0.

When the electric fields produced by a guided surface waveguide probe200 (FIG. 3) are established by matching the traveling wave phase delayof the feed network to the wave tilt angle and the probe structure isresonated with respect to the perfectly conducting image ground plane atcomplex depth z=−d/2, the fields are substantially mode-matched to aguided surface waveguide mode on the surface of the lossy conductingmedium, a guided surface traveling wave is launched along the surface ofthe lossy conducting medium. As illustrated in FIG. 1, the guided fieldstrength curve 103 of the guided electromagnetic field has acharacteristic exponential decay of e^(−αd)/√{square root over (d)} andexhibits a distinctive knee 109 on the log-log scale.

If the reactive components of the impedance seen “looking up” into thecoil and “looking down” into the lossy conducting medium are notopposite and approximately equal, then a lumped element tank circuit 260(FIG. 7C) can be included between the coil 215 (FIG. 7A) and groundstake 218 (FIG. 7A) (or grounding system). The self-resonant frequencyof the lumped element tank circuit can then be adjusted so that thereactive components “looking up” into the tank circuit of the guidedsurface waveguide probe and “looking down” into the into the lossyconducting medium are opposite and approximately equal. Under thatcondition, by adjusting the impedance (Z_(ip)) seen “looking up” intothe equivalent image plane model of FIG. 9C from the perfectlyconducting image ground plane is only resistive or Z_(ip)=R+j0.

In summary, both analytically and experimentally, the traveling wavecomponent on the structure of the guided surface waveguide probe 200 hasa phase delay (Φ) at its upper terminal that matches the angle (Ψ) ofthe wave tilt of the surface traveling wave (Φ=Ψ). Under this condition,the surface waveguide may be considered to be “mode-matched”.Furthermore, the resonant standing wave component on the structure ofthe guided surface waveguide probe 200 has a V_(MAX) at the chargeterminal T₁ and a V_(MIN) down at the image plane 139 (FIG. 8B) whereZ_(ip)=R_(ip)+j 0 at a complex depth of z=−d/2, not at the connection atthe physical boundary 136 of the lossy conducting medium 203 (FIG. 8B).Lastly, the charge terminal T₁ is of sufficient height H₁ of FIG. 3(h≥R_(x) tan ψ_(i,B)) so that electromagnetic waves incident onto thelossy conducting medium 203 at the complex Brewster angle do so out at adistance (≥R_(x)) where the 1/√{square root over (r)} term ispredominant. Receive circuits can be utilized with one or more guidedsurface waveguide probes to facilitate wireless transmission and/orpower delivery systems.

Referring back to FIG. 3, operation of a guided surface waveguide probe200 may be controlled to adjust for variations in operational conditionsassociated with the guided surface waveguide probe 200. For example, anadaptive probe control system 230 can be used to control the feednetwork 209 and/or the charge terminal T₁ to control the operation ofthe guided surface waveguide probe 200. Operational conditions caninclude, but are not limited to, variations in the characteristics ofthe lossy conducting medium 203 (e.g., conductivity σ and relativepermittivity ε_(r)), variations in field strength and/or variations inloading of the guided surface waveguide probe 200. As can be seen fromEquations (31), (41) and (42), the index of refraction (n), the complexBrewster angle (θ_(i,B)), and the wave tilt (|W|e^(jΨ)) can be affectedby changes in soil conductivity and permittivity resulting from, e.g.,weather conditions.

Equipment such as, e.g., conductivity measurement probes, permittivitysensors, ground parameter meters, field meters, current monitors and/orload receivers can be used to monitor for changes in the operationalconditions and provide information about current operational conditionsto the adaptive probe control system 230. The probe control system 230can then make one or more adjustments to the guided surface waveguideprobe 200 to maintain specified operational conditions for the guidedsurface waveguide probe 200. For instance, as the moisture andtemperature vary, the conductivity of the soil will also vary.Conductivity measurement probes and/or permittivity sensors may belocated at multiple locations around the guided surface waveguide probe200. Generally, it would be desirable to monitor the conductivity and/orpermittivity at or about the Hankel crossover distance R_(x) for theoperational frequency. Conductivity measurement probes and/orpermittivity sensors may be located at multiple locations (e.g., in eachquadrant) around the guided surface waveguide probe 200.

The conductivity measurement probes and/or permittivity sensors can beconfigured to evaluate the conductivity and/or permittivity on aperiodic basis and communicate the information to the probe controlsystem 230. The information may be communicated to the probe controlsystem 230 through a network such as, but not limited to, a LAN, WLAN,cellular network, or other appropriate wired or wireless communicationnetwork. Based upon the monitored conductivity and/or permittivity, theprobe control system 230 may evaluate the variation in the index ofrefraction (n), the complex Brewster angle (θ_(i,B)), and/or the wavetilt (|W|e^(jΨ)) and adjust the guided surface waveguide probe 200 tomaintain the phase delay (Φ) of the feed network 209 equal to the wavetilt angle (Ψ) and/or maintain resonance of the equivalent image planemodel of the guided surface waveguide probe 200. This can beaccomplished by adjusting, e.g., θ_(y), θ_(c) and/or C_(T). Forinstance, the probe control system 230 can adjust the self-capacitanceof the charge terminal T₁ and/or the phase delay (θ_(y), θ_(c)) appliedto the charge terminal T₁ to maintain the electrical launchingefficiency of the guided surface wave at or near its maximum. Forexample, the self-capacitance of the charge terminal T₁ can be varied bychanging the size of the terminal. The charge distribution can also beimproved by increasing the size of the charge terminal T₁, which canreduce the chance of an electrical discharge from the charge terminalT₁. In other embodiments, the charge terminal T₁ can include a variableinductance that can be adjusted to change the load impedance Z_(L). Thephase applied to the charge terminal T₁ can be adjusted by varying thetap position on the coil(s) 215 (FIGS. 7A-7C), and/or by including aplurality of predefined taps along the coil(s) 215 and switching betweenthe different predefined tap locations to maximize the launchingefficiency.

Field or field strength (FS) meters may also be distributed about theguided surface waveguide probe 200 to measure field strength of fieldsassociated with the guided surface wave. The field or FS meters can beconfigured to detect the field strength and/or changes in the fieldstrength (e.g., electric field strength) and communicate thatinformation to the probe control system 230. The information may becommunicated to the probe control system 230 through a network such as,but not limited to, a LAN, WLAN, cellular network, or other appropriatecommunication network. As the load and/or environmental conditionschange or vary during operation, the guided surface waveguide probe 200may be adjusted to maintain specified field strength(s) at the FS meterlocations to ensure appropriate power transmission to the receivers andthe loads they supply.

For example, the phase delay (Φ=θ_(y)+θ_(c)) applied to the chargeterminal T₁ can be adjusted to match the wave tilt angle (1). Byadjusting one or both phase delays, the guided surface waveguide probe200 can be adjusted to ensure the wave tilt corresponds to the complexBrewster angle. This can be accomplished by adjusting a tap position onthe coil(s) 215 (FIGS. 7A-7C) to change the phase delay supplied to thecharge terminal T₁. The voltage level supplied to the charge terminal T₁can also be increased or decreased to adjust the electric fieldstrength. This may be accomplished by adjusting the output voltage ofthe excitation source 212 or by adjusting or reconfiguring the feednetwork 209. For instance, the position of the tap 227 (FIG. 7A) for theexcitation source 212 can be adjusted to increase the voltage seen bythe charge terminal T₁, where the excitation source 212 comprises, forexample, an AC source as mentioned above. Maintaining field strengthlevels within predefined ranges can improve coupling by the receivers,reduce ground current losses, and avoid interference with transmissionsfrom other guided surface waveguide probes 200.

The probe control system 230 can be implemented with hardware, firmware,software executed by hardware, or a combination thereof. For example,the probe control system 230 can include processing circuitry includinga processor and a memory, both of which can be coupled to a localinterface such as, for example, a data bus with an accompanyingcontrol/address bus as can be appreciated by those with ordinary skillin the art. A probe control application may be executed by the processorto adjust the operation of the guided surface waveguide probe 200 basedupon monitored conditions. The probe control system 230 can also includeone or more network interfaces for communicating with the variousmonitoring devices. Communications can be through a network such as, butnot limited to, a LAN, WLAN, cellular network, or other appropriatecommunication network. The probe control system 230 may comprise, forexample, a computer system such as a server, desktop computer, laptop,or other system with like capability.

Referring back to the example of FIG. 5A, the complex angle trigonometryis shown for the ray optic interpretation of the incident electric field(E) of the charge terminal T₁ with a complex Brewster angle (θ_(i,B)) atthe Hankel crossover distance (R_(x)). Recall that, for a lossyconducting medium, the Brewster angle is complex and specified byequation (38). Electrically, the geometric parameters are related by theelectrical effective height (h_(eff)) of the charge terminal T₁ byEquation (39). Since both the physical height (h_(p)) and the Hankelcrossover distance (R_(x)) are real quantities, the angle of the desiredguided surface wave tilt at the Hankel crossover distance (W_(Rx)) isequal to the phase delay (Φ) of the complex effective height (h_(eff)).With the charge terminal T₁ positioned at the physical height h_(p) andexcited with a charge having the appropriate phase Φ, the resultingelectric field is incident with the lossy conducting medium boundaryinterface at the Hankel crossover distance R_(x), and at the Brewsterangle. Under these conditions, the guided surface waveguide mode can beexcited without reflection or substantially negligible reflection.

However, Equation (39) means that the physical height of the guidedsurface waveguide probe 200 can be relatively small. While this willexcite the guided surface waveguide mode, this can result in an undulylarge bound charge with little free charge. To compensate, the chargeterminal T₁ can be raised to an appropriate elevation to increase theamount of free charge. As one example rule of thumb, the charge terminalT₁ can be positioned at an elevation of about 4-5 times (or more) theeffective diameter of the charge terminal T₁. FIG. 6 illustrates theeffect of raising the charge terminal T₁ above the physical height(h_(p)) shown in FIG. 5A. The increased elevation causes the distance atwhich the wave tilt is incident with the lossy conductive medium to movebeyond the Hankel crossover point 121 (FIG. 5A). To improve coupling inthe guided surface waveguide mode, and thus provide for a greaterlaunching efficiency of the guided surface wave, a lower compensationterminal T₂ can be used to adjust the total effective height (h_(TE)) ofthe charge terminal T₁ such that the wave tilt at the Hankel crossoverdistance is at the Brewster angle.

Referring to FIG. 12, shown is an example of a guided surface waveguideprobe 200 e that includes an elevated charge terminal T₁ and a lowercompensation terminal T₂ that are arranged along a vertical axis z thatis normal to a plane presented by the lossy conducting medium 203. Inthis respect, the charge terminal T₁ is placed directly above thecompensation terminal T₂ although it is possible that some otherarrangement of two or more charge and/or compensation terminals T_(N)can be used. The guided surface waveguide probe 200 e is disposed abovea lossy conducting medium 203 according to an embodiment of the presentdisclosure. The lossy conducting medium 203 makes up Region 1 with asecond medium 206 that makes up Region 2 sharing a boundary interfacewith the lossy conducting medium 203.

The guided surface waveguide probe 200 e includes a feed network 209that couples an excitation source 212 to the charge terminal T₁ and thecompensation terminal T₂. According to various embodiments, charges Q₁and Q₂ can be imposed on the respective charge and compensationterminals T₁ and T₂, depending on the voltages applied to terminals T₁and T₂ at any given instant. I₁ is the conduction current feeding thecharge Q₁ on the charge terminal T₁ via the terminal lead, and I₂ is theconduction current feeding the charge Q₂ on the compensation terminal T₂via the terminal lead.

According to the embodiment of FIG. 12, the charge terminal T₁ ispositioned over the lossy conducting medium 203 at a physical height H₁,and the compensation terminal T₂ is positioned directly below T₁ alongthe vertical axis z at a physical height H₂, where H₂ is less than H₁.The height h of the transmission structure may be calculated as h=H₁−H₂.The charge terminal T₁ has an isolated (or self) capacitance C₁, and thecompensation terminal T₂ has an isolated (or self) capacitance C₂. Amutual capacitance C_(M) can also exist between the terminals T₁ and T₂depending on the distance therebetween. During operation, charges Q₁ andQ₂ are imposed on the charge terminal T₁ and the compensation terminalT₂, respectively, depending on the voltages applied to the chargeterminal T₁ and the compensation terminal T₂ at any given instant.

Referring next to FIG. 13, shown is a ray optics interpretation of theeffects produced by the elevated charge Q₁ on charge terminal T₁ andcompensation terminal T₂ of FIG. 12. With the charge terminal T₁elevated to a height where the ray intersects with the lossy conductivemedium at the Brewster angle at a distance greater than the Hankelcrossover point 121 as illustrated by line 163, the compensationterminal T₂ can be used to adjust h_(TE) by compensating for theincreased height. The effect of the compensation terminal T₂ is toreduce the electrical effective height of the guided surface waveguideprobe (or effectively raise the lossy medium interface) such that thewave tilt at the Hankel crossover distance is at the Brewster angle asillustrated by line 166.

The total effective height can be written as the superposition of anupper effective height (h_(UE)) associated with the charge terminal T₁and a lower effective height (h_(LE)) associated with the compensationterminal T₂ such that

$\begin{matrix}{{h_{TE} = {{h_{UE} + h_{LE}} = {{{h_{p}e^{j{({{\beta \; h_{p}} + \Phi_{U}})}}} + {h_{d}e^{j{({{\beta \; h_{d}} + \Phi_{L}})}}}} = {R_{x} \times W}}}},} & (85)\end{matrix}$

where Φ_(U) is the phase delay applied to the upper charge terminal T₁,Φ_(L) is the phase delay applied to the lower compensation terminal T₂,β=2π/λ_(p) is the propagation factor from Equation (35), h_(p) is thephysical height of the charge terminal T₁ and h_(d) is the physicalheight of the compensation terminal T₂. If extra lead lengths are takeninto consideration, they can be accounted for by adding the chargeterminal lead length z to the physical height h_(p) of the chargeterminal T₁ and the compensation terminal lead length y to the physicalheight h_(d) of the compensation terminal T₂ as shown in

h _(TE)=(h _(p) +z)e ^(j(β(h) ^(p) ^(+z)+Φ) ^(U) ⁾+(h _(d) +y)e ^(j(β(h)^(d) ^(+y)+Φ) ^(L) ₎ =R _(x) ×W.  (86)

The lower effective height can be used to adjust the total effectiveheight (h_(TE)) to equal the complex effective height (h_(eff)) of FIG.5A.

Equations (85) or (86) can be used to determine the physical height ofthe lower disk of the compensation terminal T₂ and the phase angles tofeed the terminals in order to obtain the desired wave tilt at theHankel crossover distance. For example, Equation (86) can be rewrittenas the phase delay applied to the charge terminal T₁ as a function ofthe compensation terminal height (h_(d)) to give

$\begin{matrix}{{\Phi_{U}\left( h_{d} \right)} = {{- {\beta \left( {h_{p} + z} \right)}} - {j\; {{\ln \left( \frac{{R_{x} \times W} - {\left( {h_{d} + y} \right)e^{j{({{\beta \; h_{d}} + {\beta \; y} + \Phi_{L}})}}}}{\left( {h_{p} + z} \right)} \right)}.}}}} & (87)\end{matrix}$

To determine the positioning of the compensation terminal T₂, therelationships discussed above can be utilized. First, the totaleffective height (h_(TE)) is the superposition of the complex effectiveheight (h_(UE)) of the upper charge terminal T₁ and the complexeffective height (h_(LE)) of the lower compensation terminal T₂ asexpressed in Equation (86). Next, the tangent of the angle of incidencecan be expressed geometrically as

$\begin{matrix}{{{\tan \; \psi_{E}} = \frac{h_{TE}}{R_{x}}},} & (88)\end{matrix}$

which is equal to the definition of the wave tilt, W. Finally, given thedesired Hankel crossover distance R_(x), the h_(TE) can be adjusted tomake the wave tilt of the incident ray match the complex Brewster angleat the Hankel crossover point 121. This can be accomplished by adjustingh_(p), Φ_(U), and/or h_(d).

These concepts may be better understood when discussed in the context ofan example of a guided surface waveguide probe. Referring to FIG. 14,shown is a graphical representation of an example of a guided surfacewaveguide probe 200 f including an upper charge terminal T₁ (e.g., asphere at height h_(T)) and a lower compensation terminal T₂ (e.g., adisk at height h_(d)) that are positioned along a vertical axis z thatis substantially normal to the plane presented by the lossy conductingmedium 203. During operation, charges Q₁ and Q₂ are imposed on thecharge and compensation terminals T₁ and T₂, respectively, depending onthe voltages applied to the terminals T₁ and T₂ at any given instant.

An AC source can act as the excitation source 212 for the chargeterminal T₁, which is coupled to the guided surface waveguide probe 200f through a feed network 209 comprising a phasing coil 215 such as,e.g., a helical coil. The excitation source 212 can be connected acrossa lower portion of the coil 215 through a tap 227, as shown in FIG. 14,or can be inductively coupled to the coil 215 by way of a primary coil.The coil 215 can be coupled to a ground stake (or grounding system) 218at a first end and the charge terminal T₁ at a second end. In someimplementations, the connection to the charge terminal T₁ can beadjusted using a tap 224 at the second end of the coil 215. Thecompensation terminal T₂ is positioned above and substantially parallelwith the lossy conducting medium 203 (e.g., the ground or Earth), andenergized through a tap 233 coupled to the coil 215. An ammeter 236located between the coil 215 and ground stake (or grounding system) 218can be used to provide an indication of the magnitude of the currentflow (I₀) at the base of the guided surface waveguide probe.Alternatively, a current clamp may be used around the conductor coupledto the ground stake (or grounding system) 218 to obtain an indication ofthe magnitude of the current flow (I₀).

In the example of FIG. 14, the coil 215 is coupled to a ground stake (orgrounding system) 218 at a first end and the charge terminal T₁ at asecond end via a vertical feed line conductor 221. In someimplementations, the connection to the charge terminal T₁ can beadjusted using a tap 224 at the second end of the coil 215 as shown inFIG. 14. The coil 215 can be energized at an operating frequency by theexcitation source 212 through a tap 227 at a lower portion of the coil215. In other implementations, the excitation source 212 can beinductively coupled to the coil 215 through a primary coil. Thecompensation terminal T₂ is energized through a tap 233 coupled to thecoil 215. An ammeter 236 located between the coil 215 and ground stake(or grounding system) 218 can be used to provide an indication of themagnitude of the current flow at the base of the guided surfacewaveguide probe 200 f. Alternatively, a current clamp may be used aroundthe conductor coupled to the ground stake (or grounding system) 218 toobtain an indication of the magnitude of the current flow. Thecompensation terminal T₂ is positioned above and substantially parallelwith the lossy conducting medium 203 (e.g., the ground).

In the example of FIG. 14, the connection to the charge terminal T₁ islocated on the coil 215 above the connection point of tap 233 for thecompensation terminal T₂. Such an adjustment allows an increased voltage(and thus a higher charge Q₁) to be applied to the upper charge terminalT₁. In other embodiments, the connection points for the charge terminalT₁ and the compensation terminal T₂ can be reversed. It is possible toadjust the total effective height (h_(TE)) of the guided surfacewaveguide probe 200 f to excite an electric field having a guidedsurface wave tilt at the Hankel crossover distance R_(x). The Hankelcrossover distance can also be found by equating the magnitudes ofequations (20b) and (21) for −jγρ, and solving for R_(x) as illustratedby FIG. 4. The index of refraction (n), the complex Brewster angle(θ_(i,B) and ψ_(i,B)), the wave tilt (|W|e^(jΨ)) and the complexeffective height (h_(eff)=h_(p)e^(jΦ)) can be determined as describedwith respect to Equations (41)-(44) above.

With the selected charge terminal T₁ configuration, a spherical diameter(or the effective spherical diameter) can be determined. For example, ifthe charge terminal T₁ is not configured as a sphere, then the terminalconfiguration may be modeled as a spherical capacitance having aneffective spherical diameter. The size of the charge terminal T₁ can bechosen to provide a sufficiently large surface for the charge Q₁ imposedon the terminals. In general, it is desirable to make the chargeterminal T₁ as large as practical. The size of the charge terminal T₁should be large enough to avoid ionization of the surrounding air, whichcan result in electrical discharge or sparking around the chargeterminal. To reduce the amount of bound charge on the charge terminalT₁, the desired elevation to provide free charge on the charge terminalT₁ for launching a guided surface wave should be at least 4-5 times theeffective spherical diameter above the lossy conductive medium (e.g.,the Earth). The compensation terminal T₂ can be used to adjust the totaleffective height (h_(TE)) of the guided surface waveguide probe 200 f toexcite an electric field having a guided surface wave tilt at R_(x). Thecompensation terminal T₂ can be positioned below the charge terminal T₁at h_(d)=h_(T)−h_(p), where h_(T) is the total physical height of thecharge terminal T₁. With the position of the compensation terminal T₂fixed and the phase delay Φ_(U) applied to the upper charge terminal T₁,the phase delay Φ_(L) applied to the lower compensation terminal T₂ canbe determined using the relationships of Equation (86), such that:

$\begin{matrix}{{\Phi_{U}\left( h_{d} \right)} = {{- {\beta \left( {h_{d} + y} \right)}} - {j\; {{\ln \left( \frac{{R_{x} \times W} - {\left( {h_{p} + z} \right)e^{j{({{\beta \; h_{p}} + {\beta \; z} + \Phi_{L}})}}}}{\left( {h_{d} + y} \right)} \right)}.}}}} & (89)\end{matrix}$

In alternative embodiments, the compensation terminal T₂ can bepositioned at a height h_(d) where Im{Φ_(L)}=0. This is graphicallyillustrated in FIG. 15A, which shows plots 172 and 175 of the imaginaryand real parts of Φ_(U), respectively. The compensation terminal T₂ ispositioned at a height h_(d) where Im{Φ_(U)}=0, as graphicallyillustrated in plot 172. At this fixed height, the coil phase Φ_(U) canbe determined from Re{Φ_(U)}, as graphically illustrated in plot 175.

With the excitation source 212 coupled to the coil 215 (e.g., at the 50Ωpoint to maximize coupling), the position of tap 233 may be adjusted forparallel resonance of the compensation terminal T₂ with at least aportion of the coil at the frequency of operation. FIG. 15B shows aschematic diagram of the general electrical hookup of FIG. 14 in whichV₁ is the voltage applied to the lower portion of the coil 215 from theexcitation source 212 through tap 227, V₂ is the voltage at tap 224 thatis supplied to the upper charge terminal T₁, and V₃ is the voltageapplied to the lower compensation terminal T₂ through tap 233. Theresistances R_(p) and R_(d) represent the ground return resistances ofthe charge terminal T₁ and compensation terminal T₂, respectively. Thecharge and compensation terminals T₁ and T₂ may be configured asspheres, cylinders, toroids, rings, hoods, or any other combination ofcapacitive structures. The size of the charge and compensation terminalsT₁ and T₂ can be chosen to provide a sufficiently large surface for thecharges Q₁ and Q₂ imposed on the terminals. In general, it is desirableto make the charge terminal T₁ as large as practical. The size of thecharge terminal T₁ should be large enough to avoid ionization of thesurrounding air, which can result in electrical discharge or sparkingaround the charge terminal. The self-capacitance C_(p) and C_(d) of thecharge and compensation terminals T₁ and T₂ respectively, can bedetermined using, for example, Equation (24).

As can be seen in FIG. 15B, a resonant circuit is formed by at least aportion of the inductance of the coil 215, the self-capacitance C_(d) ofthe compensation terminal T₂, and the ground return resistance R_(d)associated with the compensation terminal T₂. The parallel resonance canbe established by adjusting the voltage V₃ applied to the compensationterminal T₂ (e.g., by adjusting a tap 233 position on the coil 215) orby adjusting the height and/or size of the compensation terminal T₂ toadjust C_(d). The position of the coil tap 233 can be adjusted forparallel resonance, which will result in the ground current through theground stake (or grounding system) 218 and through the ammeter 236reaching a maximum point. After parallel resonance of the compensationterminal T₂ has been established, the position of the tap 227 for theexcitation source 212 can be adjusted to the 50Ω point on the coil 215.

Voltage V₂ from the coil 215 can be applied to the charge terminal T₁,and the position of tap 224 can be adjusted such that the phase delay(Φ) of the total effective height (h_(TE)) approximately equals theangle of the guided surface wave tilt (W_(Rx)) at the Hankel crossoverdistance (R_(x)). The position of the coil tap 224 can be adjusted untilthis operating point is reached, which results in the ground currentthrough the ammeter 236 increasing to a maximum. At this point, theresultant fields excited by the guided surface waveguide probe 200 f aresubstantially mode-matched to a guided surface waveguide mode on thesurface of the lossy conducting medium 203, resulting in the launchingof a guided surface wave along the surface of the lossy conductingmedium 203. This can be verified by measuring field strength along aradial extending from the guided surface waveguide probe 200.

Resonance of the circuit including the compensation terminal T₂ maychange with the attachment of the charge terminal T₁ and/or withadjustment of the voltage applied to the charge terminal T₁ through tap224. While adjusting the compensation terminal circuit for resonanceaids the subsequent adjustment of the charge terminal connection, it isnot necessary to establish the guided surface wave tilt (W_(Rx)) at theHankel crossover distance (R_(x)). The system may be further adjusted toimprove coupling by iteratively adjusting the position of the tap 227for the excitation source 212 to be at the 50Ω point on the coil 215 andadjusting the position of tap 233 to maximize the ground current throughthe ammeter 236. Resonance of the circuit including the compensationterminal T₂ may drift as the positions of taps 227 and 233 are adjusted,or when other components are attached to the coil 215.

In other implementations, the voltage V₂ from the coil 215 can beapplied to the charge terminal T₁, and the position of tap 233 can beadjusted such that the phase delay (Φ) of the total effective height(h_(TE)) approximately equals the angle (Ψ) of the guided surface wavetilt at R_(x). The position of the coil tap 224 can be adjusted untilthe operating point is reached, resulting in the ground current throughthe ammeter 236 substantially reaching a maximum. The resultant fieldsare substantially mode-matched to a guided surface waveguide mode on thesurface of the lossy conducting medium 203, and a guided surface wave islaunched along the surface of the lossy conducting medium 203. This canbe verified by measuring field strength along a radial extending fromthe guided surface waveguide probe 200. The system may be furtheradjusted to improve coupling by iteratively adjusting the position ofthe tap 227 for the excitation source 212 to be at the 50Ω point on thecoil 215 and adjusting the position of tap 224 and/or 233 to maximizethe ground current through the ammeter 236.

Referring back to FIG. 12, operation of a guided surface waveguide probe200 may be controlled to adjust for variations in operational conditionsassociated with the guided surface waveguide probe 200. For example, aprobe control system 230 can be used to control the feed network 209and/or positioning of the charge terminal T₁ and/or compensationterminal T₂ to control the operation of the guided surface waveguideprobe 200. Operational conditions can include, but are not limited to,variations in the characteristics of the lossy conducting medium 203(e.g., conductivity σ and relative permittivity ε_(r)), variations infield strength and/or variations in loading of the guided surfacewaveguide probe 200. As can be seen from Equations (41)-(44), the indexof refraction (n), the complex Brewster angle (θ_(i,B) and ψ_(i,B)), thewave tilt (|W|e^(jΨ)) and the complex effective height(h_(eff)=h_(p)e^(jΦ)) can be affected by changes in soil conductivityand permittivity resulting from, e.g., weather conditions.

Equipment such as, e.g., conductivity measurement probes, permittivitysensors, ground parameter meters, field meters, current monitors and/orload receivers can be used to monitor for changes in the operationalconditions and provide information about current operational conditionsto the probe control system 230. The probe control system 230 can thenmake one or more adjustments to the guided surface waveguide probe 200to maintain specified operational conditions for the guided surfacewaveguide probe 200. For instance, as the moisture and temperature vary,the conductivity of the soil will also vary. Conductivity measurementprobes and/or permittivity sensors may be located at multiple locationsaround the guided surface waveguide probe 200. Generally, it would bedesirable to monitor the conductivity and/or permittivity at or aboutthe Hankel crossover distance R_(x) for the operational frequency.Conductivity measurement probes and/or permittivity sensors may belocated at multiple locations (e.g., in each quadrant) around the guidedsurface waveguide probe 200.

With reference then to FIG. 16, shown is an example of a guided surfacewaveguide probe 200 g that includes a charge terminal T₁ and a chargeterminal T₂ that are arranged along a vertical axis z. The guidedsurface waveguide probe 200 g is disposed above a lossy conductingmedium 203, which makes up Region 1. In addition, a second medium 206shares a boundary interface with the lossy conducting medium 203 andmakes up Region 2. The charge terminals T₁ and T₂ are positioned overthe lossy conducting medium 203. The charge terminal T₁ is positioned atheight H₁, and the charge terminal T₂ is positioned directly below T,along the vertical axis z at height H₂, where H₂ is less than H₁. Theheight h of the transmission structure presented by the guided surfacewaveguide probe 200 g is h=H₁−H₂. The guided surface waveguide probe 200g includes a feed network 209 that couples an excitation source 212 suchas an AC source, for example, to the charge terminals T₁ and T₂.

The charge terminals T, and/or T₂ include a conductive mass that canhold an electrical charge, which may be sized to hold as much charge aspractically possible. The charge terminal T₁ has a self-capacitance C₁,and the charge terminal T₂ has a self-capacitance C₂, which can bedetermined using, for example, Equation (24). By virtue of the placementof the charge terminal T₁ directly above the charge terminal T₂, amutual capacitance C_(M) is created between the charge terminals T₁ andT₂. Note that the charge terminals T₁ and T₂ need not be identical, buteach can have a separate size and shape, and can include differentconducting materials. Ultimately, the field strength of a guided surfacewave launched by a guided surface waveguide probe 200 g is directlyproportional to the quantity of charge on the terminal T₁. The charge Q₁is, in turn, proportional to the self-capacitance C, associated with thecharge terminal T₁ since Q₁=C₁V, where V is the voltage imposed on thecharge terminal T₁.

When properly adjusted to operate at a predefined operating frequency,the guided surface waveguide probe 200 g generates a guided surface wavealong the surface of the lossy conducting medium 203. The excitationsource 212 can generate electrical energy at the predefined frequencythat is applied to the guided surface waveguide probe 200 g to excitethe structure. When the electromagnetic fields generated by the guidedsurface waveguide probe 200 g are substantially mode-matched with thelossy conducting medium 203, the electromagnetic fields substantiallysynthesize a wave front incident at a complex Brewster angle thatresults in little or no reflection. Thus, the surface waveguide probe200 g does not produce a radiated wave, but launches a guided surfacetraveling wave along the surface of a lossy conducting medium 203. Theenergy from the excitation source 212 can be transmitted as Zennecksurface currents to one or more receivers that are located within aneffective transmission range of the guided surface waveguide probe 200g.

One can determine asymptotes of the radial Zenneck surface currentJ_(ρ)(ρ) on the surface of the lossy conducting medium 203 to be J₁(ρ)close-in and J₂(ρ) far-out, where

$\begin{matrix}{{{{{Close}\text{-}{in}\mspace{14mu} \left( {\rho < {\lambda/8}} \right)\text{:}\mspace{14mu} {J_{\rho}(\rho)}} \sim J_{1}} = {\frac{I_{1} + I_{2}}{2{\pi\rho}} + \frac{{E_{\rho}^{QS}\left( Q_{1} \right)} + {E_{\rho}^{QS}\left( Q_{2} \right)}}{Z_{\rho}}}},{and}} & (90) \\{{{{Far}\text{-}{out}\mspace{14mu} \left( {\rho {\lambda/8}} \right)\text{:}\mspace{14mu} {J_{\rho}(\rho)}} \sim J_{2}} = {\frac{j\; {\gamma\omega}\; Q_{1}}{4} \times \sqrt{\frac{2\gamma}{\pi}} \times {\frac{e^{{- {({\alpha + {j\; \beta}})}}\rho}}{\sqrt{\rho}}.}}} & (91)\end{matrix}$

where I₁ is the conduction current feeding the charge Q₁ on the firstcharge terminal T₁, and I₂ is the conduction current feeding the chargeQ₂ on the second charge terminal T₂. The charge Q₁ on the upper chargeterminal T₁ is determined by Q₁=C₁V₁, where C₁ is the isolatedcapacitance of the charge terminal T₁. Note that there is a thirdcomponent to J₁ set forth above given by (E_(ρ) ^(Q) ¹ )/Z_(ρ), whichfollows from the Leontovich boundary condition and is the radial currentcontribution in the lossy conducting medium 203 pumped by thequasi-static field of the elevated oscillating charge on the firstcharge terminal Q₁. The quantity Z_(ρ)=jωμ_(o)/γ_(e) is the radialimpedance of the lossy conducting medium, whereγ_(e)=(jωμ₁σ₁−ω²μ₁ε₁)^(1/2).

The asymptotes representing the radial current close-in and far-out asset forth by equations (90) and (91) are complex quantities. Accordingto various embodiments, a physical surface current J(ρ) is synthesizedto match as close as possible the current asymptotes in magnitude andphase. That is to say close-in, |J(ρ)| is to be tangent to |J₁|, andfar-out |J(ρ)| is to be tangent to |J₂|. Also, according to the variousembodiments, the phase of J(ρ) should transition from the phase of J₁close-in to the phase of J₂ far-out.

In order to match the guided surface wave mode at the site oftransmission to launch a guided surface wave, the phase of the surfacecurrent |J₂| far-out should differ from the phase of the surface current|J₁| close-in by the propagation phase corresponding to e^(−jβ(ρ) ²^(−ρ) ¹ ⁾ plus a constant of approximately 45 degrees or 225 degrees.This is because there are two roots for √{square root over (γ)}, onenear π/4 and one near 5π/4. The properly adjusted synthetic radialsurface current is

$\begin{matrix}{{J_{\rho}\left( {\rho,\varphi,0} \right)} = {\frac{I_{o}\gamma}{4}{{H_{1}^{(2)}\left( {{- {jy}}\; \rho} \right)}.}}} & (92)\end{matrix}$

Note that this is consistent with equation (17). By Maxwell's equations,such a J(ρ) surface current automatically creates fields that conform to

$\begin{matrix}{{H_{\varphi} = {\frac{{- \gamma}\; I_{o}}{4}e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}}},} & (93) \\{{E_{\rho} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{u_{2}}{j\; {\omega ɛ}_{o}} \right)e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}}},{and}} & (94) \\{E_{z} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{- \gamma}{{\omega ɛ}_{o}} \right)e^{{- u_{2}}z}{{H_{0}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}.}}} & (95)\end{matrix}$

Thus, the difference in phase between the surface current |J₂| far-outand the surface current |J₁| close-in for the guided surface wave modethat is to be matched is due to the characteristics of the Hankelfunctions in equations (93)-(95), which are consistent with equations(1)-(3). It is of significance to recognize that the fields expressed byequations (1)-(6) and (17) and equations (92)-(95) have the nature of atransmission line mode bound to a lossy interface, not radiation fieldsthat are associated with groundwave propagation.

In order to obtain the appropriate voltage magnitudes and phases for agiven design of a guided surface waveguide probe 200 g at a givenlocation, an iterative approach may be used. Specifically, analysis maybe performed of a given excitation and configuration of a guided surfacewaveguide probe 200 g taking into account the feed currents to theterminals T₁ and T₂, the charges on the charge terminals T₁ and T₂, andtheir images in the lossy conducting medium 203 in order to determinethe radial surface current density generated. This process may beperformed iteratively until an optimal configuration and excitation fora given guided surface waveguide probe 200 g is determined based ondesired parameters. To aid in determining whether a given guided surfacewaveguide probe 200 g is operating at an optimal level, a guided fieldstrength curve 103 (FIG. 1) may be generated using equations (1)-(12)based on values for the conductivity of Region 1 (σ₁) and thepermittivity of Region 1 (ε₁) at the location of the guided surfacewaveguide probe 200 g. Such a guided field strength curve 103 canprovide a benchmark for operation such that measured field strengths canbe compared with the magnitudes indicated by the guided field strengthcurve 103 to determine if optimal transmission has been achieved.

In order to arrive at an optimized condition, various parametersassociated with the guided surface waveguide probe 200 g may beadjusted. One parameter that may be varied to adjust the guided surfacewaveguide probe 200 g is the height of one or both of the chargeterminals T₁ and/or T₂ relative to the surface of the lossy conductingmedium 203. In addition, the distance or spacing between the chargeterminals T₁ and T₂ may also be adjusted. In doing so, one may minimizeor otherwise alter the mutual capacitance C_(M) or any boundcapacitances between the charge terminals T₁ and T₂ and the lossyconducting medium 203 as can be appreciated. The size of the respectivecharge terminals T₁ and/or T₂ can also be adjusted. By changing the sizeof the charge terminals T₁ and/or T₂, one will alter the respectiveself-capacitances C₁ and/or C₂, and the mutual capacitance C_(M) as canbe appreciated.

Still further, another parameter that can be adjusted is the feednetwork 209 associated with the guided surface waveguide probe 200 g.This may be accomplished by adjusting the size of the inductive and/orcapacitive reactances that make up the feed network 209. For example,where such inductive reactances comprise coils, the number of turns onsuch coils may be adjusted. Ultimately, the adjustments to the feednetwork 209 can be made to alter the electrical length of the feednetwork 209, thereby affecting the voltage magnitudes and phases on thecharge terminals T₁ and T₂.

Note that the iterations of transmission performed by making the variousadjustments may be implemented by using computer models or by adjustingphysical structures as can be appreciated. By making the aboveadjustments, one can create corresponding “close-in” surface current I₁and “far-out” surface current J₂ that approximate the same currents J(ρ)of the guided surface wave mode specified in Equations (90) and (91) setforth above. In doing so, the resulting electromagnetic fields would besubstantially or approximately mode-matched to a guided surface wavemode on the surface of the lossy conducting medium 203.

While not shown in the example of FIG. 16, operation of the guidedsurface waveguide probe 200 g may be controlled to adjust for variationsin operational conditions associated with the guided surface waveguideprobe 200. For example, a probe control system 230 shown in FIG. 12 canbe used to control the feed network 209 and/or positioning and/or sizeof the charge terminals T₁ and/or T₂ to control the operation of theguided surface waveguide probe 200 g. Operational conditions caninclude, but are not limited to, variations in the characteristics ofthe lossy conducting medium 203 (e.g., conductivity σ and relativepermittivity ε_(r)), variations in field strength and/or variations inloading of the guided surface waveguide probe 200 g.

Referring now to FIG. 17, shown is an example of the guided surfacewaveguide probe 200 g of FIG. 16, denoted herein as guided surfacewaveguide probe 200 h. The guided surface waveguide probe 200 h includesthe charge terminals T₁ and T₂ that are positioned along a vertical axisz that is substantially normal to the plane presented by the lossyconducting medium 203 (e.g., the Earth). The second medium 206 is abovethe lossy conducting medium 203. The charge terminal T₁ has aself-capacitance C₁, and the charge terminal T₂ has a self-capacitanceC₂. During operation, charges Q₁ and Q₂ are imposed on the chargeterminals T₁ and T₂, respectively, depending on the voltages applied tothe charge terminals T₁ and T₂ at any given instant. A mutualcapacitance C_(M) may exist between the charge terminals T₁ and T₂depending on the distance therebetween. In addition, bound capacitancesmay exist between the respective charge terminals T₁ and T₂ and thelossy conducting medium 203 depending on the heights of the respectivecharge terminals T₁ and T₂ with respect to the lossy conducting medium203.

The guided surface waveguide probe 200 h includes a feed network 209that comprises an inductive impedance comprising a coil L_(1a) having apair of leads that are coupled to respective ones of the chargeterminals T₁ and T₂. In one embodiment, the coil L_(1a) is specified tohave an electrical length that is one-half (½) of the wavelength at theoperating frequency of the guided surface waveguide probe 200 h.

While the electrical length of the coil L_(1a) is specified asapproximately one-half (½) the wavelength at the operating frequency, itis understood that the coil L_(1a) may be specified with an electricallength at other values. According to one embodiment, the fact that thecoil L_(1a) has an electrical length of approximately one-half (½) thewavelength at the operating frequency provides for an advantage in thata maximum voltage differential is created on the charge terminals T₁ andT₂. Nonetheless, the length or diameter of the coil L_(1a) may beincreased or decreased when adjusting the guided surface waveguide probe200 h to obtain optimal excitation of a guided surface wave mode.Adjustment of the coil length may be provided by taps located at one orboth ends of the coil. In other embodiments, it may be the case that theinductive impedance is specified to have an electrical length that issignificantly less than or greater than one-half (½) the wavelength atthe operating frequency of the guided surface waveguide probe 200 h.

The excitation source 212 can be coupled to the feed network 209 by wayof magnetic coupling. Specifically, the excitation source 212 is coupledto a coil L_(P) that is inductively coupled to the coil L_(1a). This maybe done by link coupling, a tapped coil, a variable reactance, or othercoupling approach as can be appreciated. To this end, the coil L_(P)acts as a primary, and the coil L_(1a) acts as a secondary as can beappreciated.

In order to adjust the guided surface waveguide probe 200 h for thetransmission of a desired guided surface wave, the heights of therespective charge terminals T₁ and T₂ may be altered with respect to thelossy conducting medium 203 and with respect to each other. Also, thesizes of the charge terminals T₁ and T₂ may be altered. In addition, thesize of the coil L_(1a) may be altered by adding or eliminating turns orby changing some other dimension of the coil L_(1a). The coil L_(1a) canalso include one or more taps for adjusting the electrical length asshown in FIG. 17. The position of a tap connected to either chargeterminal T₁ or T₂ can also be adjusted.

Referring next to FIGS. 18A, 18B, 18C and 19, shown are examples ofgeneralized receive circuits for using the surface-guided waves inwireless power delivery systems. FIG. 18A depict a linear probe 303, andFIGS. 18B and 18C depict tuned resonators 306 a and 306 b, respectively.FIG. 19 is a magnetic coil 309 according to various embodiments of thepresent disclosure. According to various embodiments, each one of thelinear probe 303, the tuned resonators 306 a/b, and the magnetic coil309 may be employed to receive power transmitted in the form of a guidedsurface wave on the surface of a lossy conducting medium 203 accordingto various embodiments. As mentioned above, in one embodiment the lossyconducting medium 203 comprises a terrestrial medium (or Earth).

With specific reference to FIG. 18A, the open-circuit terminal voltageat the output terminals 312 of the linear probe 303 depends upon theeffective height of the linear probe 303. To this end, the terminalpoint voltage may be calculated as

V _(T)=ω₀ ^(h) ^(e) E _(inc) ·dl,  (96)

where E_(inc) is the strength of the incident electric field induced onthe linear probe 303 in Volts per meter, dl is an element of integrationalong the direction of the linear probe 303, and h_(e) is the effectiveheight of the linear probe 303. An electrical load 315 is coupled to theoutput terminals 312 through an impedance matching network 318.

When the linear probe 303 is subjected to a guided surface wave asdescribed above, a voltage is developed across the output terminals 312that may be applied to the electrical load 315 through a conjugateimpedance matching network 318 as the case may be. In order tofacilitate the flow of power to the electrical load 315, the electricalload 315 should be substantially impedance matched to the linear probe303 as will be described below.

Referring to FIG. 18B, a ground current excited coil L_(R) possessing aphase delay equal to the wave tilt of the guided surface wave includes acharge terminal T_(R) that is elevated (or suspended) above the lossyconducting medium 203. The charge terminal T_(R) has a self-capacitanceC_(R). In addition, there may also be a bound capacitance (not shown)between the charge terminal T_(R) and the lossy conducting medium 203depending on the height of the charge terminal T_(R) above the lossyconducting medium 203. The bound capacitance should preferably beminimized as much as is practicable, although this may not be entirelynecessary in every instance.

The tuned resonator 306 a also includes a receiver network comprising acoil L_(R) having a phase delay P. One end of the coil L_(R) is coupledto the charge terminal T_(R), and the other end of the coil L_(R) iscoupled to the lossy conducting medium 203. The receiver network caninclude a vertical supply line conductor that couples the coil L_(R) tothe charge terminal T_(R). To this end, the coil L_(R) (which may alsobe referred to as tuned resonator L_(R)−C_(R)) comprises aseries-adjusted resonator as the charge terminal C_(R) and the coilL_(R) are situated in series. The phase delay of the coil L_(R) can beadjusted by changing the size and/or height of the charge terminalT_(R), and/or adjusting the size of the coil L_(R) so that the phasedelay Φ of the structure is made substantially equal to the angle of thewave tilt Ψ. The phase delay of the vertical supply line can also beadjusted by, e.g., changing length of the conductor.

For example, the reactance presented by the self-capacitance C_(R) iscalculated as 1/jωC_(R). Note that the total capacitance of the tunedresonator 306 a may also include capacitance between the charge terminalT_(R) and the lossy conducting medium 203, where the total capacitanceof the tuned resonator 306 a may be calculated from both theself-capacitance C_(R) and any bound capacitance as can be appreciated.According to one embodiment, the charge terminal T_(R) may be raised toa height so as to substantially reduce or eliminate any boundcapacitance. The existence of a bound capacitance may be determined fromcapacitance measurements between the charge terminal T_(R) and the lossyconducting medium 203 as previously discussed.

The inductive reactance presented by a discrete-element coil L_(R) maybe calculated as jωL, where L is the lumped-element inductance of thecoil L_(R). If the coil L_(R) is a distributed element, its equivalentterminal-point inductive reactance may be determined by conventionalapproaches. To tune the tuned resonator 306 a, one would makeadjustments so that the phase delay is equal to the wave tilt for thepurpose of mode-matching to the surface waveguide at the frequency ofoperation. Under this condition, the receiving structure may beconsidered to be “mode-matched” with the surface waveguide. Atransformer link around the structure and/or an impedance matchingnetwork 324 may be inserted between the probe and the electrical load327 in order to couple power to the load. Inserting the impedancematching network 324 between the probe terminals 321 and the electricalload 327 can effect a conjugate-match condition for maximum powertransfer to the electrical load 327.

When placed in the presence of surface currents at the operatingfrequencies power will be delivered from the surface guided wave to theelectrical load 327. To this end, an electrical load 327 may be coupledto the tuned resonator 306 a by way of magnetic coupling, capacitivecoupling, or conductive (direct tap) coupling. The elements of thecoupling network may be lumped components or distributed elements as canbe appreciated.

In the embodiment shown in FIG. 18B, magnetic coupling is employed wherea coil L_(S) is positioned as a secondary relative to the coil L_(R)that acts as a transformer primary. The coil L_(S) may be link-coupledto the coil L_(R) by geometrically winding it around the same corestructure and adjusting the coupled magnetic flux as can be appreciated.In addition, while the tuned resonator 306 a comprises a series-tunedresonator, a parallel-tuned resonator or even a distributed-elementresonator of the appropriate phase delay may also be used.

While a receiving structure immersed in an electromagnetic field maycouple energy from the field, it can be appreciated thatpolarization-matched structures work best by maximizing the coupling,and conventional rules for probe-coupling to waveguide modes should beobserved. For example, a TE₂₀ (transverse electric mode) waveguide probemay be optimal for extracting energy from a conventional waveguideexcited in the TE₂₀ mode. Similarly, in these cases, a mode-matched andphase-matched receiving structure can be optimized for coupling powerfrom a surface-guided wave. The guided surface wave excited by a guidedsurface waveguide probe 200 on the surface of the lossy conductingmedium 203 can be considered a waveguide mode of an open waveguide.Excluding waveguide losses, the source energy can be completelyrecovered. Useful receiving structures may be E-field coupled, H-fieldcoupled, or surface-current excited.

The receiving structure can be adjusted to increase or maximize couplingwith the guided surface wave based upon the local characteristics of thelossy conducting medium 203 in the vicinity of the receiving structure.To accomplish this, the phase delay (Φ) of the receiving structure canbe adjusted to match the angle (Ψ) of the wave tilt of the surfacetraveling wave at the receiving structure. If configured appropriately,the receiving structure may then be tuned for resonance with respect tothe perfectly conducting image ground plane at complex depth z=−d/2.

For example, consider a receiving structure comprising the tunedresonator 306 a of FIG. 18B, including a coil L_(R) and a verticalsupply line connected between the coil L_(R) and a charge terminalT_(R). With the charge terminal T_(R) positioned at a defined heightabove the lossy conducting medium 203, the total phase delay Φ of thecoil L_(R) and vertical supply line can be matched with the angle (Ψ) ofthe wave tilt at the location of the tuned resonator 306 a. FromEquation (22), it can be seen that the wave tilt asymptotically passesto

$\begin{matrix}{{W = {{{W}e^{j\; \Psi}} = {\frac{E_{\rho}}{E_{z}}\underset{\rho\rightarrow\infty}{\rightarrow}\frac{1}{\sqrt{ɛ_{r} - {j\frac{\sigma_{1}}{{\omega ɛ}_{o}}}}}}}},} & (97)\end{matrix}$

where ε_(r) comprises the relative permittivity and σ_(a) is theconductivity of the lossy conducting medium 203 at the location of thereceiving structure, ε_(o) is the permittivity of free space, and ω=2πf,where f is the frequency of excitation. Thus, the wave tilt angle (Ψ)can be determined from Equation (97).

The total phase delay (Φ=θ_(c)+θ_(y)) of the tuned resonator 306 aincludes both the phase delay (θ_(c)) through the coil L_(R) and thephase delay of the vertical supply line (θ_(y)). The spatial phase delayalong the conductor length l_(w) of the vertical supply line can begiven by θ_(y)=β_(w)l_(w), where β_(w) is the propagation phase constantfor the vertical supply line conductor. The phase delay due to the coil(or helical delay line) is θ_(c)=β_(p)l_(C), with a physical length ofl_(C) and a propagation factor of

$\begin{matrix}{{\beta_{p} = {\frac{2\pi}{\lambda_{p}} = \frac{2\pi}{V_{f}\lambda_{0}}}},} & (98)\end{matrix}$

where V_(f) is the velocity factor on the structure, λ₀ is thewavelength at the supplied frequency, and λ_(p) is the propagationwavelength resulting from the velocity factor V_(f). One or both of thephase delays (θ_(c)+θ_(y)) can be adjusted to match the phase delay Φ tothe angle (Ψ) of the wave tilt. For example, a tap position may beadjusted on the coil L_(R) of FIG. 18B to adjust the coil phase delay(θ_(c)) to match the total phase delay to the wave tilt angle (Φ=Ψ). Forexample, a portion of the coil can be bypassed by the tap connection asillustrated in FIG. 18B. The vertical supply line conductor can also beconnected to the coil L_(R) via a tap, whose position on the coil may beadjusted to match the total phase delay to the angle of the wave tilt.

Once the phase delay (Φ) of the tuned resonator 306 a has been adjusted,the impedance of the charge terminal T_(R) can then be adjusted to tuneto resonance with respect to the perfectly conducting image ground planeat complex depth z=−d/2. This can be accomplished by adjusting thecapacitance of the charge terminal T₁ without changing the travelingwave phase delays of the coil L_(R) and vertical supply line. In someembodiments, a lumped element tuning circuit can be included between thelossy conducting medium 203 and the coil L_(R) to allow for resonanttuning of the tuned resonator 306 a with respect to the complex imageplane as discussed above with respect to the guided surface waveguideprobe 200. The adjustments are similar to those described with respectto FIGS. 9A-9C.

The impedance seen “looking down” into the lossy conducting medium 203to the complex image plane is given by:

Z _(in) =R _(in) +jX _(in) =Z _(o) tan h(jβ _(o)(d/2)),  (99)

where β_(o)=ω√{square root over (μ_(o)ε_(o))}. For vertically polarizedsources over the Earth, the depth of the complex image plane can begiven by:

d/2≈1/√{square root over (jωμ ₁σ₁−ω²μ₁ε₁)},  (100)

where μ₁ is the permeability of the lossy conducting medium 203 andε₁=ε_(r)ε_(o).

At the base of the tuned resonator 306 a, the impedance seen “lookingup” into the receiving structure is Z_(↑)=Z_(base) as illustrated inFIG. 9A or Z_(↑)=Z_(tuning) as illustrated in FIG. 9C. With a terminalimpedance of:

$\begin{matrix}{{Z_{R} = \frac{1}{j\; \omega \; C_{R}}},} & (101)\end{matrix}$

where C_(R) is the self-capacitance of the charge terminal T_(R), theimpedance seen “looking up” into the vertical supply line conductor ofthe tuned resonator 306 a is given by:

$\begin{matrix}{{Z_{2} = {{Z_{W}\frac{Z_{R} + {Z_{w}{\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}{Z_{w} + {Z_{R}{\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}} = {Z_{W}\frac{Z_{R} + {Z_{w}{\tanh \left( {j\; \theta_{y}} \right)}}}{Z_{w} + {Z_{R}{\tanh \left( {j\; \theta_{y}} \right)}}}}}},} & (102)\end{matrix}$

and the impedance seen “looking up” into the coil L_(R) of the tunedresonator 306 a is given by:

$\begin{matrix}{Z_{base} = {{R_{base} + {jX}_{base}} = {{Z_{R}\frac{Z_{2} + {Z_{R}{\tanh \left( {j\; \beta_{p}H} \right)}}}{Z_{R} + {Z_{2}{\tanh \left( {j\; \beta_{p}H} \right)}}}} = {Z_{c}{\frac{Z_{2} + {Z_{R}{\tanh \left( {j\; \theta_{c}} \right)}}}{Z_{R} + {Z_{2}{\tanh \left( {j\; \theta_{c}} \right)}}}.}}}}} & (103)\end{matrix}$

By matching the reactive component (X_(in)) seen “looking down” into thelossy conducting medium 203 with the reactive component (X_(base)) seen“looking up” into the tuned resonator 306 a, the coupling into theguided surface waveguide mode may be maximized.

Where a lumped element tank circuit is included at the base of the tunedresonator 306 a, the self-resonant frequency of the tank circuit can betuned to add positive or negative impedance to bring the tuned resonator306 b into standing wave resonance by matching the reactive component(X_(in)) seen “looking down” into the lossy conducting medium 203 withthe reactive component (X_(tuning)) seen “looking up” into the lumpedelement tank circuit.

Referring next to FIG. 18C, shown is an example of a tuned resonator 306b that does not include a charge terminal T_(R) at the top of thereceiving structure. In this embodiment, the tuned resonator 306 b doesnot include a vertical supply line coupled between the coil L_(R) andthe charge terminal T_(R). Thus, the total phase delay (Φ) of the tunedresonator 306 b includes only the phase delay (θ_(c)) through the coilL_(R). As with the tuned resonator 306 a of FIG. 18B, the coil phasedelay θ_(c) can be adjusted to match the angle (Ψ) of the wave tiltdetermined from Equation (97), which results in Φ=Ψ. While powerextraction is possible with the receiving structure coupled into thesurface waveguide mode, it is difficult to adjust the receivingstructure to maximize coupling with the guided surface wave without thevariable reactive load provided by the charge terminal T_(R). Includinga lumped element tank circuit at the base of the tuned resonator 306 bprovides a convenient way to bring the tuned resonator 306 b intostanding wave resonance with respect to the complex image plane.

Referring to FIG. 18D, shown is a flow chart 180 illustrating an exampleof adjusting a receiving structure to substantially mode-match to aguided surface waveguide mode on the surface of the lossy conductingmedium 203. Beginning with 181, if the receiving structure includes acharge terminal T_(R) (e.g., of the tuned resonator 306 a of FIG. 18B),then the charge terminal T_(R) is positioned at a defined height above alossy conducting medium 203 at 184. As the surface guided wave has beenestablished by a guided surface waveguide probe 200, the physical height(h_(p)) of the charge terminal T_(R) may be below that of the effectiveheight. The physical height may be selected to reduce or minimize thebound charge on the charge terminal T_(R) (e.g., four times thespherical diameter of the charge terminal). If the receiving structuredoes not include a charge terminal T_(R) (e.g., of the tuned resonator306 b of FIG. 18C), then the flow proceeds to 187.

At 187, the electrical phase delay Φ of the receiving structure ismatched to the complex wave tilt angle Ψ defined by the localcharacteristics of the lossy conducting medium 203. The phase delay(θ_(c)) of the helical coil and/or the phase delay (θ_(y)) of thevertical supply line can be adjusted to make Φ equal to the angle (Ψ) ofthe wave tilt (W). The angle (Ψ) of the wave tilt can be determined fromEquation (86). The electrical phase delay Φ can then be matched to theangle of the wave tilt. For example, the electrical phase delayΦ=θ_(c)+θ_(y) can be adjusted by varying the geometrical parameters ofthe coil L_(R) and/or the length (or height) of the vertical supply lineconductor.

Next at 190, the resonator impedance can be tuned via the load impedanceof the charge terminal T_(R) and/or the impedance of a lumped elementtank circuit to resonate the equivalent image plane model of the tunedresonator 306 a. The depth (d/2) of the conducting image ground plane139 (FIGS. 9A-9C) below the receiving structure can be determined usingEquation (100) and the values of the lossy conducting medium 203 (e.g.,the Earth) at the receiving structure, which can be locally measured.Using that complex depth, the phase shift (θ_(d)) between the imageground plane 139 and the physical boundary 136 (FIGS. 9A-9C) of thelossy conducting medium 203 can be determined using θ_(d)=β_(o) d/2. Theimpedance (Z_(in)) as seen “looking down” into the lossy conductingmedium 203 can then be determined using Equation (99). This resonancerelationship can be considered to maximize coupling with the guidedsurface waves.

Based upon the adjusted parameters of the coil L_(R) and the length ofthe vertical supply line conductor, the velocity factor, phase delay,and impedance of the coil L_(R) and vertical supply line can bedetermined. In addition, the self-capacitance (C_(R)) of the chargeterminal T_(R) can be determined using, e.g., Equation (24). Thepropagation factor (β_(p)) of the coil L_(R) can be determined usingEquation (98), and the propagation phase constant (β_(w)) for thevertical supply line can be determined using Equation (49). Using theself-capacitance and the determined values of the coil L_(R) andvertical supply line, the impedance (Z_(base)) of the tuned resonator306 as seen “looking up” into the coil L_(R) can be determined usingEquations (101), (102), and (103).

The equivalent image plane model of FIGS. 9A-9C also apply to the tunedresonator 306 a of FIG. 18B. The tuned resonator 306 a can be tuned toresonance with respect to the complex image plane by adjusting the loadimpedance Z_(R) of the charge terminal T_(R) such that the reactancecomponent X_(base) of Z_(base) cancels out the reactance component ofX_(in) of Z_(in), or X_(base)+X_(in)=0. Where the tuned resonator 306 ofFIGS. 18B and 18C includes a lumped element tank circuit, theself-resonant frequency of the parallel circuit can be adjusted suchthat the reactance component X_(tuning) of Z_(tuning) cancels out thereactance component of X_(in) of Z_(in), or X_(tuning)+X_(in)=0. Thus,the impedance at the physical boundary 136 (FIG. 9A) “looking up” intothe coil of the tuned resonator 306 is the conjugate of the impedance atthe physical boundary 136 “looking down” into the lossy conductingmedium 203. The load impedance Z_(R) can be adjusted by varying thecapacitance (C_(R)) of the charge terminal T_(R) without changing theelectrical phase delay Φ=θ_(c)+θ_(y) seen by the charge terminal T_(R).The impedance of the lumped element tank circuit can be adjusted byvarying the self-resonant frequency (f_(p)) as described with respect toFIG. 9D. An iterative approach may be taken to tune the resonatorimpedance for resonance of the equivalent image plane model with respectto the conducting image ground plane 139. In this way, the coupling ofthe electric field to a guided surface waveguide mode along the surfaceof the lossy conducting medium 203 (e.g., Earth) can be improved and/ormaximized.

Referring to FIG. 19, the magnetic coil 309 comprises a receive circuitthat is coupled through an impedance matching network 333 to anelectrical load 336. In order to facilitate reception and/or extractionof electrical power from a guided surface wave, the magnetic coil 309may be positioned so that the magnetic flux of the guided surface wave,H_(φ), passes through the magnetic coil 309, thereby inducing a currentin the magnetic coil 309 and producing a terminal point voltage at itsoutput terminals 330. The magnetic flux of the guided surface wavecoupled to a single turn coil is expressed by

=∫∫_(A) _(CS) μ_(r)μ_(o)

{circumflex over (n)}dA,  (104)

where

is the coupled magnetic flux, μ_(r) is the effective relativepermeability of the core of the magnetic coil 309, μ_(o) is thepermeability of free space,

is the incident magnetic field strength vector, {circumflex over (n)} isa unit vector normal to the cross-sectional area of the turns, andA_(CS) is the area enclosed by each loop. For an N-turn magnetic coil309 oriented for maximum coupling to an incident magnetic field that isuniform over the cross-sectional area of the magnetic coil 309, theopen-circuit induced voltage appearing at the output terminals 330 ofthe magnetic coil 309 is

$\begin{matrix}{{V = {{{- N}\frac{d\; \mathcal{F}}{dt}} \approx {{- j}\; {\omega\mu}_{r}\mu_{0}{NHA}_{CS}}}},} & (105)\end{matrix}$

where the variables are defined above. The magnetic coil 309 may betuned to the guided surface wave frequency either as a distributedresonator or with an external capacitor across its output terminals 330,as the case may be, and then impedance-matched to an external electricalload 336 through a conjugate impedance matching network 333.

Assuming that the resulting circuit presented by the magnetic coil 309and the electrical load 336 are properly adjusted and conjugateimpedance matched, via impedance matching network 333, then the currentinduced in the magnetic coil 309 may be employed to optimally power theelectrical load 336. The receive circuit presented by the magnetic coil309 provides an advantage in that it does not have to be physicallyconnected to the ground.

With reference to FIGS. 18A, 18B, 18C and 19, the receive circuitspresented by the linear probe 303, the tuned resonator 306, and themagnetic coil 309 each facilitate receiving electrical power transmittedfrom any one of the embodiments of guided surface waveguide probes 200described above. To this end, the energy received may be used to supplypower to an electrical load 315/327/336 via a conjugate matching networkas can be appreciated. This contrasts with the signals that may bereceived in a receiver that were transmitted in the form of a radiatedelectromagnetic field. Such signals have very low available power, andreceivers of such signals do not load the transmitters.

It is also characteristic of the present guided surface waves generatedusing the guided surface waveguide probes 200 described above that thereceive circuits presented by the linear probe 303, the tuned resonator306, and the magnetic coil 309 will load the excitation source 212(e.g., FIGS. 3, 12 and 16) that is applied to the guided surfacewaveguide probe 200, thereby generating the guided surface wave to whichsuch receive circuits are subjected. This reflects the fact that theguided surface wave generated by a given guided surface waveguide probe200 described above comprises a transmission line mode. By way ofcontrast, a power source that drives a radiating antenna that generatesa radiated electromagnetic wave is not loaded by the receivers,regardless of the number of receivers employed.

Thus, together one or more guided surface waveguide probes 200 and oneor more receive circuits in the form of the linear probe 303, the tunedresonator 306 a/b, and/or the magnetic coil 309 can make up a wirelessdistribution system. Given that the distance of transmission of a guidedsurface wave using a guided surface waveguide probe 200 as set forthabove depends upon the frequency, it is possible that wireless powerdistribution can be achieved across wide areas and even globally.

The conventional wireless-power transmission/distribution systemsextensively investigated today include “energy harvesting” fromradiation fields and also sensor coupling to inductive or reactivenear-fields. In contrast, the present wireless-power system does notwaste power in the form of radiation which, if not intercepted, is lostforever. Nor is the presently disclosed wireless-power system limited toextremely short ranges as with conventional mutual-reactance couplednear-field systems. The wireless-power system disclosed hereinprobe-couples to the novel surface-guided transmission line mode, whichis equivalent to delivering power to a load by a wave-guide or a loaddirectly wired to the distant power generator. Not counting the powerrequired to maintain transmission field strength plus that dissipated inthe surface waveguide, which at extremely low frequencies isinsignificant relative to the transmission losses in conventionalhigh-tension power lines at 60 Hz, all of the generator power goes onlyto the desired electrical load. When the electrical load demand isterminated, the source power generation is relatively idle.

Referring next to FIG. 20, an example of a guided surface waveguideprobe 500 is illustrated according to various embodiments of the presentdisclosure. The guided surface waveguide probe 500 is situated on aprobe site. The guided surface waveguide probe 500 is provided as anexample of the types of structures that can be used to launch guidedsurface waves on a lossy conducting media, but is not intended to belimiting or exhaustive as to those structures. Not all the structuresthat make up the guided surface waveguide probe 500 shown in FIG. 20 arenecessary in all cases, and various structures can be omitted.Similarly, the guided surface waveguide probe 500 can include otherstructures not illustrated in FIG. 20.

Among other parts, components, or structures, the guided surfacewaveguide probe 500 is constructed with a substructure 502 constructedin a lossy conducting medium 503, such as the Earth. The substructure502 forms a substructure of the guided surface waveguide probe 500 andmay be used to house various equipment as will be described. In oneembodiment, the guided surface waveguide probe 500 includes one or moreexternal phasing coils 504 and 505. The external phasing coils 504 and505 can provide both phase delay and phase shift as described below. Invarious embodiments, the external phasing coils 504 and 505 may not beused and can be omitted depending on design considerations such as thefrequency of operation and other considerations as described above.

The guided surface waveguide probe 500 can be constructed at anysuitable geographic location on the Earth. In some cases, a portion ofthe lossy conducting medium 503 around the guided surface waveguideprobe 500 can be conditioned to adjust its permittivity, conductivity,or related characteristics. The external phasing coils 504 and 505 canbe constructed at any suitable locations, including around (e.g.,encircling) the guided surface waveguide probe 500 as will be furtherdescribed below.

The substructure 502 includes a covering support slab 510 at a groundsurface elevation of the lossy conducting medium 503. To provide entryand exit points to the guided surface waveguide probe 500 forindividuals, the substructure 502 includes entryways 511 and 512,leading to staircases, for example, leading down into the substructure502. The substructure 502 also includes a number of vents 513 to exhaustforced air, for example, from heating, ventilation, and air conditioning(HVAC) systems in the substructure 502 and for potentially otherpurposes. Also, the vents 513 may be used for air intake as needed.Additionally, the substructure 502 includes an access opening 514 whichcan be used to lower various types of equipment down into thesubstructure 502.

The guided surface waveguide probe 500 includes a charge reservoir orterminal 520 (“charge terminal 520”) elevated to a height above thelossy conducting medium 503 over the substructure 502. The guidedsurface waveguide probe 500 also includes a support structure 530. Thesupport structure 530 includes a truss frame 531 and a charge terminaltruss extension 532 (“the truss extension 532”). The truss frame 531 issecured to and supported by the covering support slab 510 andsubstructure elements in the substructure 502 such as pillars and beamsas will be described.

With reference to FIG. 21, shown is a further view of the guided surfacewaveguide probe 500 according to various embodiments of the presentdisclosure. As shown, the substructure 502 is constructed to a largeextent within the lossy conducting medium 503. The substructure 502provides a supporting, foundational substructure for the guided surfacewaveguide probe 500, similar to the way a basement or cellar provides abelow-ground foundation for a building. In one example case, thesubstructure 502 can be constructed to include one floor or level at adepth of about 18 feet deep from the ground surface of the lossyconducting medium 503. In other embodiments, the substructure 502 caninclude additional underground floors and be constructed to otherdepths. Additional aspects of the substructure 502 are described belowwith reference to FIGS. 30 and 31.

The truss frame 531 includes a number of platforms supported,respectively, at elevated heights above the covering support slab 510.Among other components of the guided surface waveguide probe 500, anumber of internal phasing coil sections of the guided surface waveguideprobe 500 can be supported at one or more of the platforms as discussedin further detail below. The truss extension 532 is supported at one endby a transitional truss support region of the truss frame 531. The trussextension 532 also supports, at another end, the charge terminal 520above the lossy conducting medium 503.

To provide an example frame of reference for the size of the guidedsurface waveguide probe 500, the substructure 502 can be constructed ata size of about 92 feet in width and length, although it can beconstructed to any other suitable size. The guided surface waveguideprobe 500 can be constructed to a height of over 200 feet in oneembodiment. In that case, the charge terminal 520 can be elevated to aheight of approximately 190 feet above the lossy conducting medium 503.However, it is understood that the height of the charge terminal 520depends upon the design considerations described above, where the guidedsurface waveguide probe 500 is designed to position the charge terminal520 at a predetermined height depending on various parameters of thelossy conducting medium 503 at the site of transmission and otheroperating factors. In one example, the base of the truss frame 531 canbe constructed as a square with sides about 32 feet in length and width.It is understood that the truss frame 531 can be constructed to othershapes and dimensions. To this end, the guided surface waveguide probe500 is not limited to any particular size or dimensions and can beconstructed to any suitable size among the embodiments based on variousfactors and design considerations set forth above.

For simplicity, the truss frame 531 and the truss extension 532 of theguided surface waveguide probe 500 are drawn representatively in FIG.20. Particularly, a number of vertical, horizontal, and cross beamsupport bars of the truss frame 531 and the truss extension 532 areomitted from view in FIG. 20. Additionally, a number of gusset plates ofthe truss frame 531 and the truss extension 532 are omitted from view.The vertical, horizontal, and cross beam support bars, gusset plates,connecting hardware, and other parts of the guided surface waveguideprobe 500 are formed from non-conductive materials so as not toadversely affect the operation of the guided surface waveguide probe500. The parts of the truss frame 531 and the truss extension 532 of theguided surface waveguide probe 500 are shown in FIG. 23 and described infurther detail below.

FIG. 21 illustrates an example of the substructure 502 associated withthe guided surface waveguide probe 500 shown in FIG. 20. The lossyconducting medium 503 and sidewalls of the substructure 502 are omittedfrom view in FIG. 20. As further described below, the substructure 502includes a number of rooms or areas to store equipment, such as powertransformers, variable power and frequency power transmitters,supervisory control and data acquisition (SCADA) systems, human-machineinterface systems, electrical systems, power transmission systemmonitoring and control systems, heating, ventilation, and airconditioning (HVAC) systems, building monitoring and security systems,fire protection systems, water and air cooling systems, and othersystems. Examples of the equipment in the substructure 502 is describedin further detail below with reference to FIGS. 30 and 31.

Among a number of internal and external walls described below, thesubstructure 502 includes a foundation base 540 including a seal slab541 and a base slab 542. The seal slab 541 can be formed from pouredconcrete. According to one embodiment, the base slab 542 is also formedfrom poured concrete and is reinforced with fiberglass bars as will bedescribed.

A grounding system, which is described in further detail below withreference to FIGS. 32A and 32B, is formed and sealed in the seal slab541 of the foundation base 540. The grounding system also includes agrounding grid (not shown) in the seal slab 541, a grounding ring 551,connecting conductors 552, grounding radials 553, and other componentsnot individually referenced in FIG. 21. As described below, each of thegrounding radials 553 is electrically connected or coupled at one end tothe grounding ring 551. The other end of each of the grounding radials553 extends out from the grounding ring 551 radially away from theguided surface waveguide probe 500 to a staked location in the lossyconducting medium 503.

In one example case, the grounding radials 553 extend out about 100 feetfrom the guided surface waveguide probe 500, although other lengths ofgrounding radials 553 can be used. Further, the grounding radials 553extend out from the grounding ring 551 at a depth below the groundsurface of the lossy conducting medium 503. For example, in oneembodiment, the grounding radials 553 extend radially away from thegrounding ring 551 and the guided surface waveguide probe 500 at a depthof about 12 to 24 inches below the ground surface of the lossyconducting medium 503, although they can be buried at other depths. Thegrounding grid (not shown) in the seal slab 541, the grounding ring 551,and the grounding radials 553 provide electrical contact with the lossyconducting medium 503 for the guided surface waveguide probe 500 andvarious equipment in the substructure 502.

FIG. 22 illustrates the guided surface waveguide probe 500 shown in FIG.20 with exterior coverings 561-564 according to various embodiments ofthe present disclosure. The exterior coverings 561-564, among others,can be installed around one or both of the truss frame 531, as shown inFIG. 22, and the truss extension 532. The exterior coverings 561-564 canbe installed to insulate and protect the truss frame 531 and the trussextension 532 from the sun and various meteorological processes andevents. The exterior coverings 561-564 can also be installed tofacilitate forced-air heating and cooling of the guided surfacewaveguide probe 500 using HVAC systems, for example, installed in thesubstructure 502 or other location. Similar to the other parts of thetruss frame 531 and the truss extension 532, the exterior coverings561-564 of the guided surface waveguide probe 500 are formed fromnon-conductive materials so as not to interfere electrically with theoperation of the guided surface waveguide probe 500.

FIG. 23 illustrates an example of the support structure 530 of theguided surface waveguide probe 500. As shown in FIG. 23, the supportstructure 530 of the guided surface waveguide probe 500 can be formed asa truss, including a number of vertical, horizontal, and cross beamsupport bar members joined together using gusset plates and fasteners ata number of nodes. Cross beam support bar members, the gusset plates,and the fasteners are all nonconductive having been made fromnonconductive materials such as pultruded fiber reinforced polymer (FRP)composite structural products.

External forces on the support structure 530 primarily act at the nodes(e.g., gusset plates, fasteners) of the support structure 530 and resultin support bar member forces that are either tensile or compressive thatexert sheer forces on the gusset plates and fasteners. The supportstructure 530 is constructed so as not to exert moment forces on thegusset plates and the fasteners that form the junctions in the supportstructure 530. This accommodates the fact that the fasteners areconstructed from nonconductive materials that might have difficultywithstanding such forces without failure. The support structure 530 issecured to the covering support slab 510 using a number of base brackets565, which can be formed from metal or other appropriate material. Inone embodiment, the base brackets 565 are formed from stainless steel toreduce the possibility that the base brackets 565 would becomemagnetized.

As shown in FIG. 23, the support structure 530 includes a transitionaltruss region 570 between the truss frame 531 and the terminal trussextension 532. The transitional truss region 570 includes a number ofadditional cross beam support bars that extend and are secured betweennodes in the truss frame 531 and nodes in the terminal truss extension532. The additional cross beams in the transitional truss region 570secure the terminal truss extension 532 to the truss frame 531.

FIG. 24 illustrates a closer view of the transitional truss region 570of the guided surface waveguide probe 500, in which examples of thevertical support bars 581, the horizontal support bars 582, the crossbeam support bars 583 (collectively, “the bars 581-583), and the gussetplates 584 can be more clearly seen. As shown in FIG. 24, the trussframe 531 and the terminal truss extension 532 can be constructed usinga number of vertical support bars 581, horizontal support bars 582,cross beam support bars 583, and gusset plates 584 of various shapes andsizes. For example, the bars 581-583 can be formed as L beams, I or Hbeams, T beams, etc. at various lengths and cross-sectioned sizes. Inthat context, the bars 581-583 can be designed to translate loads to thegusset plates 584.

The gusset plates 584 can be formed as relatively thick plates ofmaterial and are used to connect a number of the bars 581-583 togetherat various nodes in the support structure 530. Each of the gusset plates584 can be fastened to a number of the bars 581-583 using nonconductivebolts or other nonconductive fastening means, or a combination offastening means. As noted above, external forces on the supportstructure 530 primarily act at the nodes gusset plates 584.

As previously mentioned, the vertical support bars 581, horizontalsupport bars 582, cross beam support bars 583, gusset plates 584,fasteners, and/or other connecting hardware, and other parts of thetruss frame 531 and the truss extension 532 can be formed (entirely orsubstantially) from non-conductive materials. For example, such supportbars 582, cross beam support bars 583, gusset plates 584, fasteners, andother connecting hardware may be constructed of pultruded fiberreinforced polymer (FRP) composite structural products. Alternatively,the same may be made out of wood or resin impregnated wood structuralproducts. In addition, other non-conductive materials may be used.

FIG. 25 is the cross-sectional view A-A of the guided surface waveguideprobe 500 designated in FIG. 20. In FIG. 25, the bars 581-583 and thegusset plates 584 of the truss frame 531 and the terminal trussextension 532 are omitted from view. Thus, among others, a number ofplatforms 591-604 of the guided surface waveguide probe 500 are shown.The platform 597 (FIG. 28) is omitted from view in FIG. 25 so as not toobscure other components of the guided surface waveguide probe 500. Theplatforms 591-593 are supported by the truss extension 532, andplatforms 594-604 are supported by the truss frame 531. In variousembodiments, individuals can access the platforms 591-604, among others,using ladders, staircases, elevators, etc. between them, as also shownin FIG. 21.

A number of additional components of the guided surface waveguide probe500 are shown in FIG. 25, including a corona hood 610 and a coil 620that, in one embodiment, can be used to inductively couple power toother electrical components of the guided surface waveguide probe 500 aswill be described. The coil 620 is supported by a coil support stand622. A power transmitter bank 630 is housed in the substructure 502.

The corona hood 610 comprises an annular canopy that tapers into a tube612. The tube 612 extends along (and through the platforms 591-596 of) aportion of the truss frame 531 and the truss extension 532 into a bottomopening of the charge terminal 520. The corona hood 610 is positionedwithin an opening in the platform 597 (FIG. 28), similar to the opening640 in the platform 598 and the other platforms 599-604. In variousembodiments, the corona hood 610 can be formed from one or moreconductive materials such as copper, aluminum, or other metal.

In one embodiment, the covering support slab 510 includes a squareopening close to its center, and the truss frame 531 is secured to thecovering support slab 510 at the base brackets 565 positioned along theperiphery of this square opening. Further, a base plate 621 can besecured over the square opening in the covering support slab 510 betweenthe covering support slab 510 and the truss frame 531. As shown, thebase plate 621 can include a circular opening in its center. The coil620 can be supported by the coil support stand 622 below, within, orabove the circular opening through the base plate 621. According to oneembodiment, the base plate 621 may be constructed of nonconductivematerials such as pultruded fiber reinforced polymer (FRP) compositestructural material and/or other nonconductive materials according toone embodiment.

In one embodiment, the external phasing coils 504 and 505 (FIG. 20) arepositioned such that at least one edge of the external phasing coils 504and/or 505 is relatively close or adjacent to the square opening in thecovering support slab 510 and the truss frame 531. In thatconfiguration, it is possible to minimize the lengths of conductorsextending between power sources in the substructure 502 and the externalphasing coils 504 and/or 505, and/or between the external phasing coils504 and/or 505 and other electrical components, such as internal phasingcoils in the tower structure of the guided surface waveguide probe 500.In addition, other openings may be created in the covering support slab510 to accommodate conductors that extend from a power source in thesubstructure 502 to one or both of the external phasing coils 504 and/or505. In one embodiment, a distance between an edge of one or both of theexternal phasing coils 504 and/or 505 and an internal phasing coilpositioned in the interior of the tower structure of the guided surfacewaveguide probe 500 is less than ⅛^(th) of the periphery of therespective coils 504 and/or 505.

The coil 620 can be embodied as a length of conductor, such as wire orpipe, for example, wrapped and supported around a coil supportstructure. The coil support structure may comprise a cylindrical body orother support structure to which the wire or pipe is attached in theform of a coil. In one example case, the coil 620 can be embodied as anumber of turns of a conductor wrapped around a support structure suchas a cylindrical housing at about 19 feet in diameter, although the coil620 can be formed to other sizes.

The power transmitter bank 630, which acts as a power source for theguided surface waveguide probe 500, is configured to convert bulk powerto a range of output power over a range of sinusoidal outputfrequencies, such as up to a megawatt of power, for example, over arange of frequencies from about 6 kHz-100 kHz, or other frequencies orfrequency ranges. As described in further detail below with reference toFIG. 30, the guided surface waveguide probe 500 can include a number ofpower transmitter cabinets, controllers, combiners, etc., such as thepower transmitter bank 630 and others. The power transmitter bank 630 isnot limited to any particular range of output power or outputfrequencies, however, as the guided surface waveguide probe 500 can beoperated at various power levels and frequencies. In one exampleembodiment, the power transmitter bank 630 comprises various componentsincluding amplifier cabinets, control cabinet, and a combiner cabinet.The amplifier cabinets may be, for example, model D120R Amplifiersmanufactured by Continental Electronics of Dallas, Tex. Likewise thecontrol cabinet and combiner cabinet are also manufactured byContinental Electronics of Dallas Tex. It is understood, however, thatpower transmitter equipment manufactured by others may be used. Inaddition, it is understood that types of power sources other than powertransmitter equipment may be used including, for example, generators orother sources.

Depending upon the operating configuration of the guided surfacewaveguide probe 500, the output of the power transmitter bank 630 (andother power transmitter banks) can be electrically coupled to the coil620. In turn, power can be inductively coupled from the powertransmitter bank 630 to other electrical components of the guidedsurface waveguide probe 500 using the coil 620. For example, power canbe inductively coupled from the coil 620 to the internal phasing coils651 shown in FIG. 26. Alternatively, one or more other coils positionedrelative (or adjacent) to the external phasing coils 504 and/or 505 canbe used to inductively couple power from the power transmitter bank 630to one or both of the external phasing coils 504 and/or 505. Forexample, such coils can be wrapped around (and supported by) the samesupport structure around which the external phasing coils 504 or 505 aresupported. In one embodiment, such coils might be placed on the groundadjacent to or below one or both of the external phasing coils 504and/or 505.

Generally, depending upon the operating frequency of the guided surfacewaveguide probe 500 (e.g., 400 Hz, 8 kHz, or 20 kHz operation), theoutput of the power transmitter bank 630 can be electrically coupled toone or more coils similar to the coil 620 for inductive coupling to oneor more internal or external phasing coils of the guided surfacewaveguide probe 500 as described herein. Additionally or alternatively,the output of the power transmitter bank 630 can be electrically coupledto one or more coils similar to the coil 620 for inductive coupling toone or more tank (inductive) coils of the guided surface waveguide probe500 as described herein.

FIG. 26 is the cross-sectional view A-A designated in FIG. 20 andillustrates a number of internal phasing coils 651 of the guided surfacewaveguide probe 500 according to various embodiments of the presentdisclosure. The internal phasing coils 651 are termed “internal” giventhat they are supported within the truss frame 531, although similarcoils can be positioned outside of the truss frame 531. Similarly, theexternal phasing coils 504 and 505 are termed “external” given that theyare placed outside of the truss frame 531.

It should be noted that the internal phasing coils 651 shown in FIG. 26are analogous to the phasing coil 215 shown in FIGS. 7A and 7B. Theinternal phasing coils 651 are also analogous to the phasing coil 215 ashown in FIG. 7C. Additionally, the external phasing coils 504 and 505are analogous to the phasing coil 215 b shown in FIG. 7C. Further, theguided surface waveguide probe 500 can include a tank circuit asdescribed below with reference to FIGS. 33A and 33B below, and thecomponents in that tank circuit are analogous to the components of thetank circuit 260 shown in FIGS. 7B and 7C.

In one embodiment, the internal phasing coils 651 are positionedadjacent to each other to create one large single internal phasing coil654. To this end, the internal phasing coils 651 may be positioned suchthat any discontinuity in the turn by turn spacing of the internalphasing coils 651 at the junction between two respective internalphasing coils 651 is minimized or eliminated, assuming that the turn byturn spacing of each of the internal phasing coils 651 is the same. Inother embodiments, the turn by turn spacing of the internal phasingcoils 651 may differ from one internal phasing coil 651 to the next. Inone embodiment, the internal phasing coils 651 may be in one or moregroups, where each group has a given turn by turn spacing.Alternatively, in another embodiment, each internal phasing coil 651 mayhave a turn by turn spacing that is unique with respect to all othersdepending on the ultimate design of the guided surface waveguide probe500. In addition, the diameters of respective ones of the internalphasing coils 651 may vary as well.

Each of the internal phasing coils 651 can be embodied as a length ofconductor, such as wire or pipe, for example, wrapped and supportedaround a support structure. In one embodiment, the support structure maycomprise a cylindrical housing or some other structural arrangement. Asone example, the internal phasing coils 651 can be about 19 feet indiameter, although other sizes can be used depending on designparameters.

The internal phasing coils 651 can be supported at one or more of theplatforms 598-604 and/or the covering support slab 510. The guidedsurface waveguide probe 500 is not limited to the use of any particularnumber of the internal phasing coils 651 or, for that matter, anyparticular number of turns of conductors in the internal phasing coils651. Instead, based on the design of the guided surface waveguide probe500, which can vary based on various operating and design factors, anysuitable number of internal phasing coils 651 can be used, where theturn by turn spacing and diameter of such internal phasing coils 651 canvary as described above.

To configure the guided surface waveguide probe 500 for use, theinternal phasing coils 651 can be individually lowered through theaccess opening 514 in the covering support slab 510, lowered into thepassageway 655, and moved through the passageway 655 to a position belowthe truss frame 531. From below the truss frame 531, the internalphasing coils 651 can be raised up into position within the openings inthe platforms 598-604 and supported at one or more of the platforms598-604. In one embodiment, each of the internal phasing coils 651 maybe hung from the structural members of a respective platform 598-604.Alternatively, each of the internal phasing coils 651 may rest onstructural members associated with a respective platform 598-604.

To raise one of the internal phasing coils 651, it can be secured to awinch line and lifted using a winch. The winch can be positioned in thetruss frame 531, the truss extension 532, and/or the charge terminal520. An example winch is shown and described below with reference toFIG. 29A. In the event that a winch is positioned in the truss frame 531or the truss extension 532, it may be attached in a temporary manner sothat the winch may be removed when necessary. In this manner, such awinch would be removeably attached to the truss frame 531 or the trussextension 532 given that such a winch would be made of conductivematerials that are likely to interfere with the operation of the guidedsurface waveguide probe 500.

In one embodiment, a conductor that extends from the bottom end of thebottom most internal phasing coil 651 is coupled to the grounding griddescribed below with reference to FIGS. 32A and 32B. Alternatively, theconductor that extends from the bottom end of the bottom most internalphasing coil 651 can be coupled to an external phasing coil, such as oneof the external phasing coils 504 and/or 505. Intermediate ones of theinternal phasing coils 651 are electrically coupled to adjacent ones ofthe internal phasing coils 651. A conductor that extends from the topend of the top most internal phasing coil 651 that is part of the singleinternal phasing coil 654 is electrically coupled to the corona hood 610and/or the charge terminal 520. If coupled to the corona hood 610, thetop most internal phasing coil 651 is coupled to the corona hood 610 ata point that is recessed up into the underside of the corona hood 610 toavoid the creation of corona as will be described.

When power is provided from the power transmitter bank 630 to the coil620 at a certain voltage and sinusoidal frequency, electrical energy istransferred from the coil 620 to the internal phasing coils 651 bymagnetic induction. To this end, the coil 620 acts as a type of primarycoil for inductive power transfer and the single internal phasing coil654 acts as a type of secondary coil. To the extent that the internalphasing coils 651 together are considered a single internal phasing coil654, then the single internal phasing coil 654 acts as the secondary. Tofacilitate magnetic induction between them, the coil 620 can bepositioned and supported by the coil support stand 622 (FIG. 25) oranother suitable structure below, within, or above the circular openingthrough the base plate 621. Further, in various cases, the coil 620 canbe positioned below, within, wholly overlapping outside, or partiallyoverlapping outside one of the internal phasing coils 651. If the coil620 is outside of the internal phasing coils 651, then the coil 620 maywholly or partially overlap a respective one of the internal phasingcoils 651. According to one embodiment, the coil 620 is positionedbelow, within, or outside a bottom most one of the internal phasingcoils 651 to facilitate a maximum charge on the charge terminal 520 asdescribed above.

To more clearly illustrate the corona hood 610, FIG. 27 is an enlargedportion of the cross-sectional view A-A designated in FIG. 20. The shapeand size of the corona hood 610 is provided as an example in FIG. 27, asother shapes and sizes are within the scope of the embodiments. Asdescribed in further detail below with reference to FIG. 27, the coronahood 610 can be positioned above and to cover at least a portion of thetop most internal phasing coil 651 (FIG. 26) in the guided surfacewaveguide probe 500. One could also say that the corona hood 610 ispositioned above and covers at least an end or top winding of the singleinternal phasing coil 654 (FIG. 26). Depending upon the number andposition of internal phasing coils 651 installed in the guided surfacewaveguide probe 500, the position of the corona hood 610 may beadjusted. Generally, the corona hood 610 can be positioned and securedat any of the platforms 594-604 of the truss frame 531. However, theposition of the corona hood 610 generally needs to be at a sufficientheight so as not to create an unacceptable amount of bound capacitancein accordance with the discussion above. If necessary, sections of thetube 612 can be installed (or removed) to adjust the position of thecorona hood 610 to one of the platforms 594-604.

The corona hood 610 is designed to minimize or reduce atmosphericdischarge around the conductors of the end windings of the top-mostinternal phasing coil 651. To this end, atmospheric discharge may occuras Trichel pulses, corona, and/or a Townsend discharge. The Townsenddischarge may also be called avalanche discharge. All of these differenttypes of atmospheric discharges represent wasted energy in thatelectrical energy flows into the atmosphere around the electricalcomponent causing the discharge to no effect. As the voltage on aconductor is continually raised from low voltage potential to highvoltage potential, atmospheric discharge may manifest itself first asTrichel pulses, then as corona, and finally as a Townsend discharge.Corona discharge in particular essentially occurs when current flowsfrom a conductor node at high potential, into a neutral fluid such asair, ionizing the fluid and creating a region of plasma. Coronadischarge and Townsend discharges often form at sharp corners, points,and edges of metal surfaces. Thus, to reduce the formation ofatmospheric discharges from the corona hood 610, the corona hood 610 isdesigned to be relatively free from sharp corners, points, edges, etc.

To this end, the corona hood 610 terminates along an edge 611 thatcurves around in a smooth arc and ultimately is pointed toward theunderside of the corona hood 610. The corona hood 610 is an invertedbowl-like structure having a recessed interior that forms a hollow 656in the underside of the corona hood 610. An outer surface 657 of thebowl-like structure curves around in the smooth arc mentioned above suchthat the edge of the bowl-like structure is pointed toward the recessedinterior surface 658 of the hollow 656.

During operation of the guided surface waveguide probe 500, the chargedensity on the outer surface 657 of the corona hood 610 is relativelyhigh as compared to the charge density on the recessed interior surface658 of the corona hood 610. As a consequence, the electric fieldexperienced within the hollow 656 bounded by the recessed interiorsurface 658 of the corona hood 610 will be relatively small as comparedto the electric field experienced near the outer surface 657 of thecorona hood 610. According to the various embodiments, the end mostwindings of the top-most internal phasing coil 651 are recessed into thehollow 656 bounded by the recessed interior surface 658 of the coronahood 610. Given that the electric fields in the hollow 656 arerelatively low, atmospheric discharge is prevented or at least minimizedfrom conductors recessed into the hollow 656. Specifically, in thisarrangement, atmospheric discharge is prevented or minimized from theend most windings of the top-most internal phasing coil 651 that arerecessed into the hollow 656. Also, atmospheric discharge is preventedfrom forming or minimized from the lead that extends from the end mostwinding of the top-most internal phasing coil 651 to an attachment pointon the recessed interior surface 658 of the corona hood 610. Thus, bypositioning the corona hood 610 such that the top winding(s) of thehighest most internal phasing coil 651 is recessed into the hollow 656having lower electric fields, atmospheric discharge is prevented fromforming or is minimized around the top winding and the lead extendingfrom the top winding which experience the highest electrical potentialof the entire system.

The corona hood 610 terminates by tapering into a tube 612 that extendsfrom the corona hood 610 to the charge terminal 520. The tube 612 actsas a conductor between the corona hood 610 and the charge terminal 520and includes one or more bends or turns 614 from the corona hood 610 tothe charge terminal 520. In the case of the guided surface waveguideprobe 500, the turn 614 is relied upon to shift the tube 612 to anoff-center position within the platforms 591-593, among others, in thetruss extension 532. In that way, space can be reserved on the platforms591-593 for individuals to stand and service the guided surfacewaveguide probe 500. The tube 612 may include a pivot junction above theturn 614 that would allow the tube 612 to be swung out of position overthe corona hood 610 to leave an open hole in the tube 612 or the taperedportion of the corona hood 610 just above the corona hood 610. This isdone to allow a cable to pass through the center of the corona hood 610to facilitate lifting coil sections into place as described herein.Alternatively, a portion of the tube 612 may be removeable at the firstbend of the turn 614 to allow a cable to pass through the center of thecorona hood 619.

Given that the corona hood 610 and the tube 612 are formed from aconductive material, the highest-installed internal coil 651 can beelectrically coupled to the corona hood 610 by connecting the top mostwinding to the corona hood 610 at a point on the recessed interiorsurface 658 of the corona hood 610 to prevent atmospheric discharge fromoccurring around the connection point as well as the lead extending fromthe top most winding to the connection point on the recessed interiorsurface 658 of the corona hood 610. Alternatively, if such atmosphericdischarge is not prevented entirely, then it is at least minimized inorder to minimize unwanted losses. In that case, the conductor can beelectrically coupled to the recessed interior surface 658 of the coronahood 610 at a point where the corona hood 610 tapers into the tube 612,for example, or at any other suitable location.

FIG. 28 is a cross-sectional view of the charge terminal 520 of theguided surface waveguide probe 500 shown in FIG. 20. The charge terminal520 is positioned at the top of the guided surface waveguide probe 500above the truss extension 532. Individuals can access the interior spacewithin the charge terminal 520 using ladders 660 and 661, among others,to reach the top platform 670 of the truss extension 532. The topplatform 670 includes an opening 671 through which a winch line canpass. As described in further detail below with reference to FIGS. 29Aand 29B, a winch can be used to raise one or more of the internalphasing coils 651 into place, so that they can be secured at one or moreof the platforms 598-604 (FIG. 25).

The charge terminal 520 can be formed from any suitable conductive metalor metals, or other conductive materials, to serve as a charge reservoirfor the guided surface waveguide probe 500. As shown, the chargeterminal 520 includes a hollow hemisphere portion 680 at the top thattransitions into a hollow toroid portion 681 at the bottom. The hollowtoroid portion 681 turns to the inside of the charge terminal 520 andends at an annular ring lip 682.

For an electrical connection to the internal phasing coils 651, the tube612 can extend further up toward the top of the charge terminal 520. Asshown in the inset in FIG. 28, one or more coupling conductors 690,formed from a conductive material, can extend radially away from the topof the tube 612. The coupling conductors 690 can be mechanically andelectrically coupled to any point on the inner surface of the chargeterminal 520. For example, the coupling conductors 690 can beelectrically and mechanically connected to points around the annularring lip 682. Alternatively, the coupling conductors 690 can bemechanically and electrically coupled to points on the inside surface ofthe hollow toroid portion 681 or the hollow hemisphere portion 680. Thecharge terminal 520 is generally attached to and supported by the trussextension 532 as described below with reference to FIGS. 29A and 29B.

FIGS. 29A and 29B illustrate top and bottom perspective views,respectively, of a top support platform 700 of the guided surfacewaveguide probe 500 shown in FIG. 20 according to various embodiments ofthe present disclosure. In the example of the guided surface waveguideprobe 500 described and illustrated herein, the charge terminal 520shown in FIG. 28 can surround the top support platform 700.

The top support platform 700 is supported at the top of the trussextension 532 of the guided surface waveguide probe 500. Similar to thebars 581-583 referenced in FIG. 24, the truss extension 532 includes anumber of vertical support bars 710, horizontal support bars 711, andcross beam support bars 712. The truss extension 532 also includes anumber of gusset plates 713 to secure the vertical support bars 710,horizontal support bars 711, and cross beam support bars 712 together.

Secured at the top of the truss extension 532, the top support platform700 includes a mounting ring 720 as shown in FIG. 29B. In oneembodiment, the annular ring lip 682 of the charge terminal 520 can besecured to the mounting ring 720 using bolts or other suitable hardware.In that way, the charge terminal 520 can be mounted to the top supportplatform 700, which is secured to the truss extension 532.

The top support platform 700 includes an arrangement of platform joists730 and a railing 731. The top platform 670 (FIG. 28) can be seated uponand secured to the platform joists 730. The top support platform 700also includes a winch 740. The winch 740 can be used to install,reconfigure, and maintain various components of the guided surfacewaveguide probe 500. For example, a winch line of the winch 740 can berouted through the top support platform 700, through the opening 671(FIG. 28) in the top platform 670, and down into the truss extension 532and the truss frame 531. The winch line can be lowered down toward andinto the passageway 655 (FIG. 26) in the substructure 502 (FIG. 26).From there, the winch line can be secured to one of the internal phasingcoils 651 (FIG. 27), and the internal phasing coil 651 can be lifted upinto the truss frame 531 and secured. Given that the winch 740 islocated inside the charge terminal 520, the winch 740 is located in theregion of uniform electric potential and is safe from discharge, eddycurrents, or interference. In order to power the winch 740, anelectrical cord may be brought up to the winch 740 from a power sourcesuch as utility power when the guided surface waveguide probe 500 is notoperational. During operation, however, such an electrical cord would beremoved.

The components of the top support platform 700, including the verticalsupport bars 710, horizontal support bars 711, cross beam support bars712, gusset plates 713, platform joists 730, railing 731, etc. may beformed (entirely or substantially) from non-conductive materials.Alternatively, the same may be formed from conductive materials sincethey are located in a region of uniform electrical potential. In anyevent, such components may be constructed from lightweight materialssuch as aluminum or titanium so as to reduce the physical load on theentire structure of the guided surface waveguide probe 500.

FIGS. 30 and 31 illustrate various components inside the substructure502 of the guided surface waveguide probe 500 shown in FIG. 20 accordingto various embodiments of the present disclosure. The arrangement of therooms, compartments, sections, stairwells, etc., in the substructure 502is provided as a representative example in FIGS. 30 and 31. In otherembodiments, the space within the substructure 502 can be configured foruse in any suitable way, and the equipment described below can beinstalled in various locations.

The substructure 502 includes external walls 800 and internal walls 801.According to one embodiment, the external walls 800 and internal walls801 are formed from poured concrete and, in some cases, reinforced withfiberglass rebar as will be described. For safety, the internal walls801 can be designed at a suitable thickness and/or structural integrityto withstand or retard the spread of fire, coronal discharge, etc.Various entryways and passages through the internal walls 801 permitindividuals and equipment to move throughout the substructure 502. Theentryways and passages can be sealed using any suitable types of doors,including standard doors, sliding doors, overhead doors, etc. As alsoshown, a pathway 802 is reserved through various areas in thesubstructure 502 for individuals to walk around and install, service,and move the equipment in the substructure 502, as necessary.

A number of the pillars 810, not all of which are individuallyreferenced in FIG. 30, support the covering support slab 510 (FIG. 20)of the guided surface waveguide probe 500. The pillars 810 can be formedfrom reinforced concrete or other suitable materials as will bedescribed. A central group of the pillars 810 are positioned under eachof the base brackets 565 to support the truss frame 531 and the rest ofthe structure.

Stairwells 820 and 821 are provided at opposite corners of thesubstructure 502. The stairwells 820 and 821 lead up to the entryways511 and 512 (FIG. 20). The stairwell 820 is surrounded by a stairwellenclosure 822, but stairwell enclosures are not necessary in every case.For example, the stairwell 821 is not shown as being enclosed in FIG.30. The enclosure around each stairwell 820 and 821 provides for safetyin case of fire or other calamity. Also, the stairwell enclosure 822prevents or retards the entry of water into the substructure 502.

The substructure 502 includes a number of different rooms, compartments,or sections separated by the internal walls 801. Various types ofequipment is installed in the rooms or compartments of the substructure502. Among other types of equipment and systems, a power transmitterbanks 630 and 631, a motor controller 830, a number of transformers 831,and an HVAC system 832 can be installed in the substructure 502 as shownin FIG. 30. Further, as shown in FIG. 31, a supervisory control and dataacquisition (SCADA) system 840, an arc flash detection system 841, and afire protection system 842 can be installed in the substructure 502.Additionally, although not referenced in FIGS. 30 and 31, an electricalswitching gear can be installed in the substructure 502 to receive powerover one or more power transmission cables 850 and connect the power tothe transformers 831 and, in turn, other equipment in the substructure502.

In one embodiment, the power transmitter bank 630 can be embodied as anumber of variable power, variable frequency, power transmitters capableof outputting power over a range of sinusoidal output frequencies, suchas up to a megawatt of power, for example, over a range of frequenciesfrom about 6 kHz-100 kHz. However, the power transmitter bank 630 canprovide output power at lower and higher wattages and at lower andhigher frequencies in various embodiments. The power transmitter banks630 and 631 are examples of various power sources that may be used suchas, for example, generators and other power sources. The powertransmitter bank 630 includes a control cabinet 632, a combiner 633, anda number of power transmitters 634. Each of the power transmitters 634can include a number of power amplifier boards, and the outputs of thepower transmitters 634 can be tied or combined together in the combiner633 before being fed to the coil 620 (FIG. 25) of the guided surfacewaveguide probe 500, for example. The second power transmitter bank 631is similar in form and function as the power transmitter bank 630.

Depending upon the operating configuration of the guided surfacewaveguide probe 500, the output of the power transmitter banks 630 and631 can be electrically coupled to the coil 620 within the substructure502, where the coil 620 acts as a primary coil to inductively coupleelectrical energy into the internal phasing coils 651. Alternatively,the output of the power transmitter banks 630 and 631 may be coupled tocoils acting as primaries that are positioned around the externalphasing coils 504 and 505, or the inductive coil 263/942 (FIG. 7C/FIGS.33A and B) as described herein. Thus, electrical energy may be appliedto the guided surface waveguide probe 500 by way of inductive couplingfrom a coil acting as a primary to any one of the internal phasing coils651, the external phasing coils 504/505, or inductive coils 263/942.

In one embodiment, power can be fed from the power transmission cables850 at a voltage level for power transmission at 138 kV (or higher), atthe voltage level for sub-transmission at 26 kV or 69 kV, at the voltagelevel for primary customers at 13 kV or 4 kV, at the voltage level forinternal customers at 120V, 240V, or 480V, or at another suitablevoltage level.

The power can be fed through electrical switch gear and to thetransformers 831. The electrical switch gear can include a number ofrelays, breakers, switchgears, etc., to control (e.g., connect anddisconnect) the connection of power from the cables 850 to the equipmentinside the substructure 502. The power can be fed from the transformers831, at a stepped-up or stepped-down voltage, to the power transmitterbanks 630 and 631. Alternatively, the power transmitter banks 630 and631 can be supplied directly with power at a suitable voltage, such as480V or 4160V, for example, from the cables 850.

The motor controller 830 can control a number of forced air and waterheating and/or cooling subsystems in the substructure 502, among othersubsystems. To this end, various ducts and piping are employed to routecooling air and water to various locations and components of the guidedsurface waveguide probe 500 to prevent damage to the system andstructure due to heat. The SCADA system 840 can be relied upon tomonitor and control equipment in the guided surface waveguide probe 500,such as the power transmitter banks 630 and 631, motor controller 830,transformers 831, HVAC system 832, arc flash detection system 841, andfire protection system 842, among others.

In one embodiment, the entire substructure 502 including the foundationbase 540, seal slab 541, external walls 800, internal walls 801, pillars810, and the covering support slab 510 (FIG. 20) is formed using pouredconcrete reinforced with Glass Fiber Reinforced Polymer (GFRP) rebar.The concrete used may include an additive that reduces the amount ofmoisture in the cement to reduce the conductivity of the cement toprevent eddy currents and the like in the cement itself. In oneembodiment, such an additive may comprise XYPEX™ manufactured by XypexChemical Corporation of Richmond, British Columbia, Canada, or otherappropriate additive. The GFRP rebar ensures that there are noconductive pathways in the cement upon which eddy currents might beproduced.

FIGS. 32A and 32B illustrate the grounding system 900 of the guidedsurface waveguide probe 500 shown in FIG. 20. The grounding system 900includes a grounding grid 910, the grounding ring 551, connectingconductors 552, a number of grounding radials 553, and a number ofground stakes 920. The grounding system 900 is shown as a representativeexample in FIGS. 32A and 32B and can differ in size, shape, andconfiguration in other embodiments. The grounding system 900 can beformed from conductive materials and provides an electrical connectionto the lossy conducting medium 503 (e.g., the Earth) for the guidedsurface waveguide probe 500 and the equipment in the substructure 502.

In one embodiment, the grounding grid 910 is surrounded in the seal slab541 of the foundation base 540 (FIG. 21). The grounding system 900 alsoincludes a number of grounding stakes 920 driven into the lossyconducting medium 503 below the grounding grid 910 and electricallycoupled to the grounding grid 910.

The connecting conductors 552 extend from the grounding grid 910 to thegrounding ring 551. The grounding radials 553 are electrically coupledat one end to the grounding ring 551 and extend out from the groundingring 551 radially away from the guided surface waveguide probe 500 to anumber of grounding stakes 920 driven into the lossy conducting medium503. The grounding ring 551 includes an opening or break 930 to preventcirculating current in the grounding ring 551 itself. Together all ofthe grounding components of the grounding system 900 provide a pathwayfor current generated by the guided surface waveguide probe 500 to thelossy conducting medium 503 around the guided surface waveguide probe500.

FIG. 33A illustrates an example tank circuit 940 a of the guided surfacewaveguide probe 500 according to various embodiments of the presentdisclosure. The tank circuit 940 a includes an inductive coil 942, anumber of parallel capacitors 944A-944D, and a number of switches946A-946D corresponding to the parallel capacitors 944A-944D. Withreference to the tank circuit 260 shown in FIGS. 7B and 7C, theinductive coil 942 is analogous to the inductive coil 263 and theparallel capacitors 944A-944D are analogous to the capacitor 266. Notethat although only a limited number of capacitors are shown, it isunderstood that any number of capacitors may be employed and switchedinto the tank circuit 940 a as conditions demand.

The tank circuit 940 a can be electrically coupled at one end as shownin FIG. 33A to one or more phasing coils, such as the single internalphasing coil 654, the external phasing coils 504 and/or 505, and/orother phasing coils. The tank circuit 940 a can be electrically coupledat another end as shown in FIG. 33A to a grounding system, such as thegrounding system 900 shown in FIGS. 32A and 32B.

The capacitors 944A-944D can be embodied as any suitable type ofcapacitor and each can store the same or different amounts of charge invarious embodiments, for flexibility. Any of the capacitors 944A-944Dcan be electrically coupled into the tank circuit 940 a by closingcorresponding ones of the switches 946A-946D. Similarly, any of thecapacitors 944A-944D can be electrically isolated from the tank circuit940 a by opening corresponding ones of the switches 946A-946D. Thus, thecapacitors 944A-944D and the switches 946A-946D can be considered a typeof variable capacitor with a variable capacitance depending upon whichof the switches 946A-946D are open (and closed). Thus, the equivalentparallel capacitance of the parallel capacitors 944A-944D will dependupon the state of the switches 946A-946D, thereby effectively forming avariable capacitor.

The inductive coil 942 can be embodied as a length of conductor, such aswire or pipe, for example, wrapped and supported around a coil supportstructure. The coil support structure may comprise a cylindrical body orother support structure to which the wire or pipe is attached in theform of a coil. In some cases, the connection from the inductive coil942 to the grounding system 900 can be adjusted using one or more taps943 of the inductive coil 942 as shown in FIG. 7A. Such a tap 943 maycomprise, for example, a roller or other structure to facilitate easyadjustment. Alternatively, multiple taps 943 may be employed to vary thesize of the inductive coil 942, where one of the taps 943 is connectedto the capacitors 944.

As described herein, a phasing coil such as the single internal phasingcoil 654 and the external phasing coils 504 and 505 can provide bothphase delay and phase shift. Further, the tank circuit 940 a thatincludes the inductive coil 942 can provide a phase shift without aphase delay. In this sense, the inductive coil 942 comprises a lumpedelement assumed to have a uniformly distributed current throughout. Inthis respect, the inductive coil 942 is electrically small enoughrelative to the wavelength of transmission of the guided surfacewaveguide probe 500 such that any delay it introduces is relativelynegligible. That is to say, the inductive coil 942 acts as a lumpedelement as part of the tank circuit 940 a that provides an appreciablephase shift, without a phase delay.

FIG. 33B illustrates another example tank circuit 940 b of the guidedsurface waveguide probe 500 according to various embodiments of thepresent disclosure. As compared to the tank circuit 940 a shown in FIG.33A, the tank circuit 940 b includes a variable capacitor 950 in placeof the capacitors 944A-944D and switches 946A-946D. With reference tothe tank circuit 260 shown in FIGS. 7B and 7C, the inductive coil 942 isanalogous to the inductive coil 263 and the variable capacitor 950 isanalogous to the capacitor 266.

As shown, the variable capacitor 950 can be buried or embedded into thelossy conducting medium 503, such as the Earth. The variable capacitor950 includes a pair of cylindrical, parallel charge conductors 952, 954and an actuator 960. The actuator 960, which can be embodied as ahydraulic actuator that actuates a hydraulic piston. Alternatively, theactuator 960 may be embodied as an electric actuator that employs amotor or other electrical component that drives a screw shaft or othermechanical lifting structure. Further, the actuator 960 may be embodiedas a pneumatic actuator that is employed to raise or lower a pneumaticcylinder. Still other types of actuators may be employed to move theinner charge conductor 952 relative to the outer charge conductor 954,or vice versa, or both. Also, some other type of actuator may beemployed beyond those described herein.

The actuator 960 is configured to raise and lower the inner chargeconductor 952 within, or relative to, the outer charge conductor 954. Byraising and lowering the inner charge plate 952 with respect to theouter charge plate 954, the capacitance of the variable capacitor 950can be modified and, thus, the electrical characteristics of the tankcircuit 940 b adjusted.

While the variable capacitor 950 is shown as being buried in the lossyconducting medium 503, it is understood that the variable capacitor 950may also reside in a building or a substructure such as the substructure502. Also, while the variable capacitor 950 is depicted as beingcylindrical in shape, it is possible to use any shape such asrectangular, polygonal, or other shape.

As discussed above, atmospheric discharge can occur as Trichel pulses,corona, and/or a Townsend discharge. A Townsend discharge can also becalled avalanche discharge. All of these different types of atmosphericdischarges represent wasted energy in that electrical energy flows intothe atmosphere around the electrical component causing the discharge tono effect. As the voltage on a conductor is continually raised from lowvoltage potential to high voltage potential, atmospheric discharge canmanifest itself first as Trichel pulses, then as corona, and finally asa Townsend discharge. Corona discharge in particular essentially occurswhen the current flows from a conductor node at high potential, into aneutral fluid such as air, ionizing the fluid and creating a region ofplasma. Corona discharge and Townsend discharges often form at sharpcorners, points, and edges of metal surfaces.

As can be appreciated, various components of the guided surfacewaveguide probe 500 can be subject to atmospheric discharge due to theincreasing voltage associated with the operation of the guided surfacewaveguide probe 500. For example, atmospheric discharge can occur at thecharge terminal 520, the tube 612, the internal coil 654, and the baseof the primary coil 620. However, the likelihood of atmosphericdischarge at the bottom of the primary coil 620 will be naturallyminimized due to low voltage level as compared to the voltage levels atthe other areas.

According to various embodiments of the present disclosure, the amountof atmospheric discharge can be minimized to a nominal amount accordingto one or more factors, such as, for example, the use of a corona hood510, the use of corona rings, the effective diameter of the internalcoil 654, the effective diameter of the tube 612, the size and shape ofthe charge terminal 520, and/or other factors that can be appreciated.The nominal amount can be defined as a level of atmospheric dischargethat is below a threshold corresponding to the amount of atmosphericdischarge where energy loss effects the operation and/or efficiency ofthe guided surface waveguide probe 520, the amount of atmosphericdischarge where damage to the components and/or structure of the guidedsurface waveguide probe 500 can occur, and/or any other factor as can beappreciated.

Moving on to FIG. 34, shown is an example of a cross-sectional view ofthe charge terminal 520 of the guided surface waveguide probe 500according to various embodiments of the present disclosure. As shown,the charge terminal 520 includes a hollow hemisphere portion 680 at thetop that transitions into a hollow toroid portion 681 at the bottom. Thehollow toroid portion 681 turns to the inside of the charge terminal 520and ends at an annular ring lip 682. Similarly to the corona hood 520,the charge terminal 520 is shaped so that there are no sharp corners,points, and edges, which are prone to causing atmospheric discharge.

The charge terminal 520 further comprises an exterior surface 1008 and arecessed interior surface 1010. The recessed interior surface 1010defines a charge terminal hollow 1012. Similar to the corona hood 620,the shape of the charge terminal 520 is designed such that atmosphericdischarge is prevented and/or minimized. Specifically, during operationof the guided surface waveguide probe 500, the charge density on theexterior surface 1008 of the charge terminal 520 is relatively high incomparison to the charge density on the recessed interior surface 1010of the charge terminal 520. As a consequence, the electric fieldexperienced within the charge terminal hollow 1012 bounded by therecessed interior surface 1010 of the charge terminal 520 will berelatively small as compared to the electric field experienced near theexterior surface 1008 of the charge terminal 520.

Turning now to FIG. 35, shown is an example of a portion of anotherembodiment of the guided surface waveguide probe 500 according tovarious embodiments of the present disclosure. Specifically, in someembodiments, the internal phasing coil 654 can be inductively coupled tothe charge terminal 520, thereby eliminating the corona hood 610 and thefeed tube 612. As such, FIG. 35 illustrates and example of the internalphasing coil 654 recessed within the charge terminal region 1012 of thecharge terminal 520. As further shown in FIG. 35, the guided surfacewaveguide probe can include a tank circuit 260 which can be can beelectrically coupled at one end to one or more phasing coils, such asthe single internal phasing coil 654, the external phasing coils 504and/or 505, and/or other phasing coils.

According to various embodiment, the shape of the charge terminal 510can minimize the amount of atmospheric discharge to a nominal amount.Specifically, during operation of the guided surface waveguide probe500, the charge density on the outer surface of the charge terminal 510is relatively high as compared to the charge density on the recessedinterior surface 1010 of the corona hood charge terminal 510. As aconsequence, the electric field experienced within the hollow region1012 bounded by the recessed interior surface 1010 of the chargeterminal 520 will be relatively small as compared to the electric fieldexperienced near the exterior surface 1008 of the charge terminal 610.

According to various embodiments, a top portion 1002 of the interiorphasing coil 654 that contains the end most windings of the highest mostinternal phasing coil 651 is recessed within the charge terminal 520.Given that the electric fields in the hollow region 1012 are relativelylow, atmospheric discharge is prevented or at least minimized fromconductors recessed within the hollow region 1012. Specifically, in thisarrangement, atmospheric discharge is prevented or minimized from theend most windings of the top-most internal phasing coil 651 that arerecessed within the hollow region 1012.

Also, atmospheric discharge is prevented from forming or is minimizedfrom the lead 1004 that extends from the end-most winding of thetop-most phasing coil 651 to an attachment point 1006 on the recessedinterior surface 1010 of the charge terminal 520. Thus, by positioningthe charge terminal 520 such that the top-portion 1002 of the highestmost internal phasing coil 651 is recessed into the hollow region 1012having lower electric fields, atmospheric discharge is prevented fromforming or is minimized around the top winding and the lead 1004extending from the top winding which experiences the highest electricalpotential of the entire system. It should be noted that while FIG. 35illustrates a single lead 1004 and single attachment point 1006, theinternal phasing coil 654 can be coupled to the recessed internalsurface 1010 via multiple leads 1004 and attachment points 1006. The topwinding of the highest internal phasing coil 651 can be mechanically andelectrically coupled to any point on the recessed interior surface 1010of the charge terminal 520. For example, in some embodiments, theinternal phasing coil 654 can be attached via solder, mechanicalconnectors, welding, and/or any other suitable form of attachment as canbe appreciated.

Turning now to FIG. 36, shown is another embodiment of a portion of theguided surface waveguide probe according to various embodiments of thepresent disclosure. Similarly to FIG. 35, the internal phasing coil 654extends into the hollow region 1010 of the charge terminal 520. However,FIG. 36 differs from FIG. 35 in that the top portion 1016 of theinternal phasing coil 654 tapers into a conical coil. It should be notedthat the shape of the internal phasing coil 54 is not limited to acylinder, and can comprise a conical shape, a cylindrical pyramid, arectangular pyramid, an exponential pyramid, and/or any other acceptableshape as can be appreciated. As such, the shape of the internal phasingcoil 654 should not be limited to a cylindrical from. However, it shouldbe noted that the likelihood of atmospheric discharge is dependent uponthe effective diameter and/or radius of curvature of the internalphasing coil 654.

Specifically, as the effective diameter and/or width of a conductorincreases, the equipotential surface of the conductor increases allowinga greater amount of voltage that can be applied prior to causingbreakdown, i.e., atmospheric discharge. The radius of curvature is thereciprocal of curvature and corresponds to the curve at a particularpoint of an arc. The curvature of the circle is the reciprocal of theradius of the circle, and relates to how curved something is in relationto a flat plane. As such, as the effective diameter increases, theradius of curvature increases. This results in an increase of theequipotential surface of the conductor as well as an increase in theamount of voltage that can be applied prior to the occurrence ofatmospheric discharge. As such, the effective diameter of both theinternal phasing coil 654 must be sized large enough to eliminate orreduce to a nominal amount of atmospheric discharge. It should be notedthat while the internal phasing coil 654 comprises a spiral structure,as voltage rises with every turn, the surface of the internal phasingcoil 654 can begin to behave as a cylinder which can create theatmospheric discharge concerns, as can be appreciated.

With respect to the internal phasing coil 654 of FIG. 36, the taperedportion would need to be recessed within the hollow region 1012 tomaintain reduction of atmospheric discharge. As the coil 654 begins totaper, the equipotential surface begins to decrease, which can cause abreakdown resulting in atmospheric discharge. Due to the shape of thecharge terminal 520, the electric field experienced within the hollowregion 1012 bounded by the recessed interior surface 1010 of the chargeterminal 520 will be relatively small as compared to the electric fieldexperienced near the exterior surface 1008 of the charge terminal 610.As such, the tapering of the top portion 1016 of the internal phasingcoil 654 will not result in substantial atmospheric discharge so long asthe tapered portion is recessed within the hollow region 1012.

Disjunctive language such as the phrase “at least one of X, Y, or Z,”unless specifically stated otherwise, is otherwise understood with thecontext as used in general to present that an item, term, etc., may beeither X, Y, or Z, or any combination thereof (e.g., X, Y, and/or Z).Thus, such disjunctive language is not generally intended to, and shouldnot, imply that certain embodiments require at least one of X, at leastone of Y, or at least one of Z to each be present.

It should be emphasized that the above-described embodiments of thepresent disclosure are merely possible examples of implementations setforth for a clear understanding of the principles of the disclosure.Many variations and modifications may be made to the above-describedembodiment(s) without departing substantially from the spirit andprinciples of the disclosure. All such modifications and variations areintended to be included herein within the scope of this disclosure andprotected by the following claims. In addition, all optional andpreferred features and modifications of the described embodiments anddependent claims are usable in all aspects of the disclosure taughtherein. Furthermore, the individual features of the dependent claims, aswell as all optional and preferred features and modifications of thedescribed embodiments are combinable and interchangeable with oneanother.

Therefore, the following is claimed:
 1. A guided surface waveguideprobe, comprising: a charge terminal elevated over a lossy conductingmedium, the charge terminal being configured to minimize atmosphericdischarge; a coil being configured to provide a voltage to the chargeterminal with a phase delay (Φ) that matches a wave tilt angle (Ψ)associated with a complex Brewster angle of incidence (θ_(i,B))associated with the lossy conducting medium, a top portion of the coilbeing recessed within a hollow region of the charge terminal; and anexcitation source coupled to the coil.
 2. The guided surface waveguideprobe of claim 1, wherein the charge terminal comprises an invertedbowl-like structure having a hemispherical portion at a top thattransitions into a hollow toroid portion at a bottom, the hollow toroidportion terminating at an annular lip that points upwardly towards anunderside of the charge terminal.
 3. The guided surface waveguide probeof claim 1, wherein the charge terminal comprises an exterior surfaceand a recessed interior surface, the recessed interior surface definingthe hollow region.
 4. The apparatus of claim 3, wherein a charge densityon the exterior surface of the charge terminal is greater than a chargedensity on the recessed interior surface.
 5. The guided surfacewaveguide probe of claim 1, wherein the atmospheric discharge of thecoil is minimized due to the top portion of the coil being recessedwithin the hollow region.
 6. The guided surface waveguide probe of claim1, wherein the charge terminal is substantially free of any one of: asharp edge, a sharp corner, or a sharp point.
 7. The guided surfacewaveguide probe of claim 1, wherein an electric field within the hollowregion is substantially smaller than the electric field experienced nearan exterior surface of the charge terminal.
 8. The apparatus of claim 1,wherein the atmospheric discharge comprises at least one of: Trichelpulses, corona, or a Townsend discharge.
 9. The apparatus of claim 1,wherein the top portion of the coil comprises a conical shape.
 10. Anapparatus, comprising: a charge terminal elevated over a lossyconducting medium, the charge terminal comprising an inverted bowl-likestructure having a hemispherical portion at a top that transitions intoa hollow toroid portion at a bottom, the hollow toroid portionterminating at an annular lip that points upwardly towards an undersideof the charge terminal; and a coil configured to provide a voltage tothe charge terminal with a phase delay (Φ) that matches a wave tiltangle (Ψ) associated with a complex Brewster angle of incidence(θ_(i,B)) associated with the lossy conducting medium, a top portion ofthe coil being recessed within a hollow region of the charge terminal.11. The apparatus of claim 10, wherein atmospheric discharge associatedwith the coil is minimized due to the top portion being recessed withinthe hollow region.
 12. The apparatus of claim 11, wherein theatmospheric discharge comprises at least one of: Trichel pulses, corona,or a Townsend discharge.
 13. The apparatus of claim 10, wherein aneffective diameter of the coil is sized such that an amount ofatmospheric discharge associated with the coil is below a thresholdcorresponding to an acceptable level of atmospheric discharge.
 14. Theapparatus of claim 10, wherein the coil comprises a conical shape, acylindrical shape, a cylindrical pyramid shape, a rectangular pyramidshape, or an exponential pyramid shape.
 15. The apparatus of claim 10,wherein a charge density on an exterior surface of the charge terminalis greater than a charge density on a recessed interior surface of thecharge terminal.
 16. The apparatus of claim 10, wherein an electricfield within the hollow region is substantially smaller than theelectric field experienced near an exterior surface of the chargeterminal.
 17. A guided surface waveguide probe, comprising: a chargeterminal elevated over a lossy conducting medium, the charge terminalbeing shaped to reduce atmospheric discharge; a coil coupled to anexcitation source and configured to provide a voltage to the chargeterminal with a phase delay (Φ) that matches a wave tilt angle (Ψ)associated with a complex Brewster angle of incidence (θ_(i,B))associated with the lossy conducting medium, a top portion of the coilbeing recessed within a hollow region of the charge terminal, and aneffective diameter of at least a portion of the coil being sized suchthat atmospheric discharge associated with an increase of voltage isminimized.
 18. The guided surface waveguide probe of claim 17, whereinthe charge terminal comprises an inverted bowl-like structure having ahemispherical portion at a top that transitions into a hollow toroidportion at a bottom, the hollow toroid portion terminating at an annularlip that points upwardly towards an underside of the charge terminal, arecessed interior surface of the charge terminal defining the hollowregion.
 19. The guided surface waveguide probe of claim 17, wherein anelectric field within the hollow region is substantially smaller thanthe electric field experienced near an exterior surface of the chargeterminal.
 20. The guided surface waveguide probe of claim 17, whereinthe coil is part of a feed network, and the feed network furthercomprises a lumped element tank circuit comprising an inductor and avariable capacitor connected in parallel.